Hi Tibees. I have watched some of your videos with interest but this one made me a subscriber. As a dad of three young girls I think you are a great role model. If you could do some videos in this style at the primary and secondary level that would be fantastic. Just a thought....
Foxlady, 4 ironic comments you don't deserve, and 566 voices do well approve your opinion. You are right : this girl is wholesome, and this makes most of viewers feeling good.
Something about the trees, the sunlight, and the birds chirping is just truly peaceful. The setting behind the content really made this more beautiful and calming overall.
We, the impoverished faceless mass of TH-cam audience, hereby respectfully yet firmly demand more of such content for the redemption of our collective soul in these dark, ignorant, and brutal times. Here’s a new sub and future patron. A Mathematical day to you!
You are truly amazing. Like Feynman, you can take difficult concepts and, because you truly understand them, make them seem so simple that virtually anyone could understand them. You have a rare and precious gift. Thank you for these videos!!
What you said, that's why our schools are failing children. What you describe as her genius? That's really caring a great deal. It can't be bought or paid for and from want of it many students myself included have suffered aversion to learning certain subjects.
Yes that is what I thought of too. I am reading Feynman book The Strange Theory or Light and Matter and it has a certain way of explaining concepts that is rare to come by.
I love your presentation Tibees!! When written a little differently, Euler's Identity appears as even more awesome. e^πi + 1 = 0 This simple identity establishes a relationship between the 5 most important constants and the 3 fundamental operations, all in one single equation, with nothing in excess and nothing lacking. One can never tire of pondering this identity; it is like contemplating a bouquet of the 5 most beautiful flowers in Mathematics, arranged with extraordinary simplicity and elegance.
Melancholy Blacklite lmao, my math teacher is like robot, sits still, talks small and only repeat the definition and no explanation. We always jump right in solving equations..... 😭😭😭😭😭😭
And if sin jumps on top of cos, they produce a tan baby. And when the tan baby turns upside down it produces a cot for it to sleep in (which is useless since the baby is now the cot)
After thousands of videos about e^{iπ} I finally could understand the proof of it. Thank you so much!!! Being an enthusiastic of mathematics is a lot easier with you. English is not even my native language but the simplicity of this video totally overcame the linguistic barrier. Keep what you're doing girl, you're awesome :)
This was absolutely mesmerizing, I have never thought math could be relaxing. If only..... ah well, high school would have been so different. Thanks for this!
When I first saw this demonstrated, the professor started with the definition that the derivative of e^x equals e^x and developed the Taylor series from that. I remember that it "blew my mind"!
I love to draw and I am an undergraduate physics student. Lately I was feeling a little demotivated about my course and this really lifted my spirits up! Thank you so much for your efforts! Love ya! ❤️❤️❤️
It's amazing how even after 11 years of learning this at school, I'm still able to follow what you're saying. I remember how crazy it used to be solving mathematical problems especially calculus problems given the highly competitive exams that we have in India. But after so many years, I wonder what use or benefit all this has other than academic benefits by getting good scores.
The great outdoors is likely where many of the ancients taught their pupils. I'm sure it made for calm and inspiring atmosphere in which to teach. We really should hold school outside on nice days.
if you take the Taylor series of e^(ix) you get cos(x) + i*sin(x) now if u plug pi in there u get a creampie xDD just kidding had to write that. anyways if you plug pi in there you simply get degrees with radians, which are calculated with pi(not sure about the math there, but that's how it is) so cos(pi) + i*(sin(pi))= -1. put it in a calculator and u will get -1. in fact, if you simply put cos(pi) + sin(pi) you will also get -1.
@@marxk4rl well, that's the beauty of it, if you plus pi into sine or cosine you get an integer, and you can cancel out the imaginary number too. for example i*i=-1. now if you studied calculus you'd know you can represent sine and cosine by series, and this is exactly what you get with e^(i*pi) you get the representation of cos(pi) + isin(pi) so the math and it will give u the same result as i*i.
Wish I had this kind of enthusiasm for mathematics. As a 3rd year student (towards Electronics and Electricity Engineering) I do from time to time find myself amazed by the beauty of it. Really liked the way you approached that proof, if only my teachers could show it to me the way you did.
IDK if it's the beauty of mathematics or the beauty of Toby's demonstration or her own beauty that's making me go crazy.Probably a mixture of all three! Gonna come back to these vids for sure
Thanks Toby...I was surrounded by university level science oriented people growing up, got my degree in math, but somehow never ran across this. Very neat interrelation of four special numbers, e=mc² like!
This one is solution of euler's form. Cos(a)+i Sin(a) =e^ia i=iota, put a=π.. And get the most famous result e^iπ=-1... Actually we use to study it in complex no. Of high school(11th stndrd)
This is the first time I understand, without over romanticizing in my head, the real beauty of an equation. Pure beautiful mathematical knowledge. You are really amazing. Thank you, my brain is really happy right now!.
I don't think people get the intuition behind this as well; any real number to the power of i represents a rotation on the unit circle on the argand plane, and when that real base is e, the rotation on the argand plane gives exactly one radian anti-clockwise from 1+0i. Multiplying by pi would therefore mean 1 rad * pi = 180 degrees thus it would land you at a full 180 degree rotation from 1+0i to -1+0i. Then you add 1, meaning a translation of one unit to the right which gives 0+0i. Thus e^(ipi) + 1 = 0
That’s quite nice. I think it was GH Hardy who said that this equations should be obvious to any mathematician. Your intuitive explanation is the closest I’ve got to understanding that statement. In case others were wondering, the Argand plane is the complex plane (x = real, y = imaginary). Also, 180 degrees = pi radians.
Toby, thanks for this lovely video. My heart skipped a beat after calmly following your explanation and then suddenly recognizing the pattern in the series for cos(x) and sin(x) in exp ix 😉
I love your passion for the science. You are my Idol scientific. I'm studying chemistry, but all into science is beautiful. Greetings from Mexico. Tibees ✨🌟 😊
I have watched this a few times and really like the way toby presents this. I think a quite hard headed approach is required for mathematical content, and Toby contrasts this with her smiling personality.
This is without a doubt the best channel on TH-cam. I could listen to you explain math and physics all day. It reinforces all the things I learned while pursuing my engineering degree and it actually is so much fun lol
I loved Bob Ross so much. He was my first and best art teacher. I think of him and talk about with my wife often. She now likes to watch the old Bob Ross videos too. This is still all totally over my head though sadly. But for the first time ever, I feel like I could learn maths from someone. You seem right and the Bob Ross link is an ethereal helper. I'm going to look through your videos and see if there's a good entry point..... Thank you for doing these. I love that someone out there thinks a bit like I do! I had to school myself alone for reasons too detailed to go into but I still do the same thing. I take my ignorance and use it as a measuring tool to find the correct shape and size of a subject....
I had a Math teacher in College that scared me and my classmates during one year in College. When i was playing during break with my friends, when the ring bell to stop the break we we're all scared in front of our classroom door. Everybody was happy to go in class, except us. And i hated math because of her. Gladly i did economy and sociology with help of other teachers and now i'm a teacher too in economy and sociology but it was difficult (because of maths and using statistics). What i'm saying is... Your video prove Math can be teaching in a soft, calm and cheesy atmosphere. I think we could learn to do Maths during spring or summer , outside, around natural environment like you did. It's amazing, congrats about your video.
This video is proof that even mathematics can be used to heal stress, not just music. By the way, when you said don't be scared by the equations on the board, i was greatly, " relieved " 😊😊😂😂😂
The proof is elegant and simple once the series expansion is taken for granted. Please do videos for series expansion proofs. Thanks so much. Never understood it my college days, its really simple , elegant and clear explanation.
For the first time in this life I could see that e^(i*pi) = -1 has three distinct constants all related by this equation. Thank you for making a fantastic video. I really like the scenic backdrop. It uplifts the mood immediately. I saw a Bob Ross video as well. That was also a treat.
Actually we arn’t going to need many normal math teachers.. we will just have these famous online ones that are permanent features for humanity. They make learning their lovely subjects so fun & alive. The future humans will be so smart!!! Yay
Here's the perfect equation : (beauty + intelligence = Tibees) I hear so much of your interest in psychology that you mentioned in a different video coming through in the way you explain things. I love that.🙌
The taylor series, sequences, and recursive formulas are a thing of beauty. Whoever came up with this algebra to generalize an infinite sum into a general formula with finite concepts and to see if it converges to a finite value or infinite value is just marvelous. I especially like how you can add k+1 to prove your results as well. What is your favorite equation?
This video is inspired by a Bob Ross tutorial: th-cam.com/video/lLWEXRAnQd0/w-d-xo.html
"The uploader has not made this video available in your country." :(
(Not your video, the Bob Ross video)
How professional
Great video imaginary friend (i) is great makes the equation more interesting .
This is beautiful indeed.
Hi Tibees. I have watched some of your videos with interest but this one made me a subscriber. As a dad of three young girls I think you are a great role model. If you could do some videos in this style at the primary and secondary level that would be fantastic. Just a thought....
I’ve never been so calm while watch someone do mathematics before.
Billy bansi calming magical voice. And she’s always smiling
Quantum ASMR has some physics/maths videos that are super calming
i agree. same think while watching video.
Please explain graph of exponential function
this is the single most wholesome girl in existence
How do you know? Have you ever measured them all?
@@TegaraMusic Confirmed!
Foxlady, 4 ironic comments you don't deserve, and 566 voices do well approve your opinion. You are right : this girl is wholesome, and this makes most of viewers feeling good.
@Mohan Sandal are you an Indian??
yo can we report this incel indian guy to TH-cam, pls?
Toby, I swear you‘re barely a step away from becoming a meme.
Should I be concerned?
Wouldn't that be fulfillment of life purpose?
is that a good thing????
who knows ?
@@tibees
Yes.Albeit an inspirational meme. Like the success kid meme.
Voice plus the accent are so satisfying and calming
WARNING: High levels of charm.
Charmed, I am. Great teacher!!!
Charm is a characteristic of Psychopathy. Beware of overly charming people.
misread charm
@@markcaseon7136 sigma male here
Something about the trees, the sunlight, and the birds chirping is just truly peaceful. The setting behind the content really made this more beautiful and calming overall.
Some happy little integrals.
Hahahahahahahha
Oh my god🤣😂😂
because everyone needs a friend
We, the impoverished faceless mass of TH-cam audience, hereby respectfully yet firmly demand more of such content for the redemption of our collective soul in these dark, ignorant, and brutal times.
Here’s a new sub and future patron.
A Mathematical day to you!
lol stoop
@@DTB1752 Stoop?
You are truly amazing. Like Feynman, you can take difficult concepts and, because you truly understand them, make them seem so simple that virtually anyone could understand them. You have a rare and precious gift. Thank you for these videos!!
What you said, that's why our schools are failing children. What you describe as her genius? That's really caring a great deal. It can't be bought or paid for and from want of it many students myself included have suffered aversion to learning certain subjects.
Yes that is what I thought of too. I am reading Feynman book The Strange Theory or Light and Matter and it has a certain way of explaining concepts that is rare to come by.
Agreed
"in mathematics, we don't make mistakes, we have happy accidents."
...that cost us a degree
It explains why I always got C: I'm too happily mistaken.
Reggie87 well not my math teacher tho
Yeah these happy accidents costs us money and lectures from our parents.
I used to get happy Cs
The most tender mathematical demonstration I've seen in my life haha Love your videos!
This is my new favorite thing.
Now you know better about math
I like your channel too
Next video: Why math is so underrated....
I just find this, is wonderful!!!
"no mistakes here, just happy accidents" - you said it so naturally. I adore these videos.
Tell that any NASA scientist lmao
@@petersall1055 lol touché
Bob
Courtesy: Brilliant dot organization.
Stares
I love your presentation Tibees!! When written a little differently, Euler's Identity appears as even more awesome. e^πi + 1 = 0 This simple identity establishes a relationship between the 5 most important constants and the 3 fundamental operations, all in one single equation, with nothing in excess and nothing lacking. One can never tire of pondering this identity; it is like contemplating a bouquet of the 5 most beautiful flowers in Mathematics, arranged with extraordinary simplicity and elegance.
Where have you been all my life!
Hi :)
Woah Cody amazed to see you here👍🏻
wow, such an honor, watching your channel mins ago as well!
@@tibees if this comment said by common people then u just ignored it
@@vishaljeekri7752 who wouldn’t want to say hi to a god?
“I think we’re going to have a great time. So let’s get into it”
I wish my math teacher could be like this...
Melancholy Blacklite lmao, my math teacher is like robot, sits still, talks small and only repeat the definition and no explanation. We always jump right in solving equations..... 😭😭😭😭😭😭
and certainly never any "don't worry, I'll be right there with you". She was always "that hag that give me bad grades out of disappointment"
Mine was exactly like this but she lied. I never had a great time. Just homework......My God the homework....
Well this was way more soothing than my video...great work!
MajorPrep mainly her voice😁
PREP YOU WATCH HER TOO!? LOVE BOTH YOURE VIDEOS!
@@crimsonseraph3543 wait was he called majorprep before?????
wait yeah i just saw his video about changing it
this sine function looks a little lonely, lets make him a little cosine friend
Gr8
And if sin jumps on top of cos, they produce a tan baby.
And when the tan baby turns upside down it produces a cot for it to sleep in (which is useless since the baby is now the cot)
Oldest joke in the math book
Never heard of it..
Her explanation is amazing! In addition to that her patience and smile is just incredible! Good job!
Thank you for making it so understandable. Your soothing voice also helps.
After thousands of videos about e^{iπ} I finally could understand the proof of it. Thank you so much!!!
Being an enthusiastic of mathematics is a lot easier with you.
English is not even my native language but the simplicity of this video totally overcame the linguistic barrier.
Keep what you're doing girl, you're awesome :)
"Hold on to your calculators, ladies and gentlemen!"
Toby said it first!
Yup, 4:01 ... I'm not even sure why I loved that bit, but I did. :)
This is cheesy in all the right ways
This was absolutely mesmerizing, I have never thought math could be relaxing. If only..... ah well, high school would have been so different. Thanks for this!
If i had a teacher like you... soft spoken and motivational (saying math is no headache)
perhaps I could have had a credit in it.
Please make your channel into this. This was extremely therapeutic
Beautifully explained! The birds chirping behide was amazing! Thank you
When I first saw this demonstrated, the professor started with the definition that the derivative of e^x equals e^x and developed the Taylor series from that. I remember that it "blew my mind"!
you have a very peaceful way of presenting math that makes it delightful.
I love to draw and I am an undergraduate physics student. Lately I was feeling a little demotivated about my course and this really lifted my spirits up! Thank you so much for your efforts! Love ya! ❤️❤️❤️
You should do a TED - I am sure it would be a best one
um she already has
Brenda Benjamin I have not done a TED talk. That's a wish for the future
Tibees I'm sorry I got confused with that presentation you did but I forget what it was about. I could have sworn it was a TED talk.
Please, DON'T.
It's amazing how even after 11 years of learning this at school, I'm still able to follow what you're saying. I remember how crazy it used to be solving mathematical problems especially calculus problems given the highly competitive exams that we have in India.
But after so many years, I wonder what use or benefit all this has other than academic benefits by getting good scores.
And that preytel is the proverbial Onion!
The great outdoors is likely where many of the ancients taught their pupils.
I'm sure it made for calm and inspiring atmosphere in which to teach.
We really should hold school outside on nice days.
Loved watching this, in fact I've watched it a few times. I don't think I've seen this formula proved before. Thank you.
While watching this video, there's definitely the "ups" time in my life. Thank you.
Looking foward to the next episode of the joy of mathematics
Nature makes no mistakes, just happy accidents
I heard that somewhere also... Who said that? :D
here. this equation might have a little friend living nearby... .
@@Saidor570 bob ross
Oh yeah, it was so obvious! I feel stupid now! Thanks :D
@@Saidor570 Don't worry, people make no mistakes Just happy accidents
It's fun when YOU describe and explain it all. You are a natural teacher.
"No mistakes here, just happy accidents."
love that mindset.
how do you get -1 from e^ipi?
you beat the imagination out of it.
Ehh.. I think beating imagination just gets you "pie". I'm not complaining though.
@@neehalramchund2643You just get cream. You must plug it into a function first.
if you take the Taylor series of e^(ix) you get cos(x) + i*sin(x) now if u plug pi in there u get a creampie xDD just kidding had to write that.
anyways if you plug pi in there you simply get degrees with radians, which are calculated with pi(not sure about the math there, but that's how it is) so cos(pi) + i*(sin(pi))= -1.
put it in a calculator and u will get -1. in fact, if you simply put cos(pi) + sin(pi) you will also get -1.
@@invorokner282 I don't think an irrational/imaginary number can be equal to an integer. It's like saying the infinite is a number.
@@marxk4rl well, that's the beauty of it, if you plus pi into sine or cosine you get an integer, and you can cancel out the imaginary number too. for example i*i=-1.
now if you studied calculus you'd know you can represent sine and cosine by series, and this is exactly what you get with e^(i*pi) you get the representation of cos(pi) + isin(pi) so the math and it will give u the same result as i*i.
But, why would anyone dislike this? This was so beautiful! ❤️
Wondering the same thing here...
You are so gentle and graceful....
Wish I had this kind of enthusiasm for mathematics. As a 3rd year student (towards Electronics and Electricity Engineering) I do from time to time find myself amazed by the beauty of it. Really liked the way you approached that proof, if only my teachers could show it to me the way you did.
IDK if it's the beauty of mathematics or the beauty of Toby's demonstration or her own beauty that's making me go crazy.Probably a mixture of all three!
Gonna come back to these vids for sure
I cried to this
Could you do more series like this?
+1
Good pun
Taylor ones you mean?
divergent series plz
I think it's worth considering the form *e^i(pi) + 1 = 0*
which has:
e, i, pi, 1, and 0
addition, multiplication, and exponentiation
It is your world. You can do whatever you want to do in it.
:-)
also, 0 is another beautiful number :)
Toby you're my idol. This video absolutely rules, i am at peace.
Why you stopped making awesome videos?
'My love for mathematics'... I have literally not smiled this broadly in years.
Thanks Toby...I was surrounded by university level science oriented people growing up, got my degree in math, but somehow never ran across this. Very neat interrelation of four special numbers, e=mc² like!
I love this video: simple and beautiful.
I look forward to seeing more mathematical art :)
I could listen to this lovely young lady read the friggen phone book!
You actually have me watching videos on math. You're very good at explaining things and I can tell you enjoy it. Thanks for the videos!
People like Toby prove that, in what can be a terrible world, there are careful angels doing quiet work. May God richly bless you.
This one is solution of euler's form. Cos(a)+i Sin(a) =e^ia
i=iota, put a=π.. And get the most famous result e^iπ=-1... Actually we use to study it in complex no. Of high school(11th stndrd)
Heard it from somewhere,
i said: let's be rational,
pi said: let's get real,
and e said to both: Join me and we'll be -one!
also pi: We could be rational, or we could be TRANSCENDENTAL!
😠 you're choking the joke.
I feel left out!
@MorTobXD The original went like this...
i: be rational!
pi: get real!
Brilliant
What page was i just recommended...
This is gold.
This is freaking brilliant. Let's build a happy little cos. 😊
Pure ASMR. What a voice. What a pronunciation. Delightful.
This is the first time I understand, without over romanticizing in my head, the real beauty of an equation. Pure beautiful mathematical knowledge. You are really amazing. Thank you, my brain is really happy right now!.
This is the best 'not understanding something' i've ever had..
This amazing lady exists on a level that I cannot even imagine. Much respect for you skills.
This is superb! ASMR mathematics. I just subscribed :-)
Thanks
Whoa!!!I Am addicted to this channel. That smooth voice, disarming smile, sure makes me a great learner. Natural gifted teacher. Thank you!!!
Thank you for this tutorial and your inspiring explanation.
I don't think people get the intuition behind this as well; any real number to the power of i represents a rotation on the unit circle on the argand plane, and when that real base is e, the rotation on the argand plane gives exactly one radian anti-clockwise from 1+0i. Multiplying by pi would therefore mean 1 rad * pi = 180 degrees thus it would land you at a full 180 degree rotation from 1+0i to -1+0i. Then you add 1, meaning a translation of one unit to the right which gives 0+0i. Thus e^(ipi) + 1 = 0
Wizard Koer I’ve got a belly button. 😀
That’s quite nice. I think it was GH Hardy who said that this equations should be obvious to any mathematician. Your intuitive explanation is the closest I’ve got to understanding that statement.
In case others were wondering, the Argand plane is the complex plane (x = real, y = imaginary). Also, 180 degrees = pi radians.
Toby, thanks for this lovely video. My heart skipped a beat after calmly following your explanation and then suddenly recognizing the pattern in the series for cos(x) and sin(x) in exp ix 😉
I love your passion for the science. You are my Idol scientific. I'm studying chemistry, but all into science is beautiful. Greetings from Mexico. Tibees ✨🌟 😊
I have watched this a few times and really like the way toby presents this. I think a quite hard headed approach is required for mathematical content, and Toby contrasts this with her smiling personality.
The world would be so much better if we all had had teachers like her in school 😪
I feel like this vid is missing a couple of dragons
Lol, I like that..!! 😂
I thought the same Game of thrones 😁
Please make more of these. I loved it.
i love your dog, such a well behaved doggo
😺
That is the best and most amazing way that I have seen someone prove this equation ever!! Thank you and great job
This is without a doubt the best channel on TH-cam. I could listen to you explain math and physics all day. It reinforces all the things I learned while pursuing my engineering degree and it actually is so much fun lol
This was absolutely brilliant! I've never thought I could enjoy watching a math video. Thank you :)
Tibees says that she has an imaginary friend
Start thinking she's a loon...
Tibees writes "i"
Nevermind 🤣
I loved Bob Ross so much. He was my first and best art teacher. I think of him and talk about with my wife often. She now likes to watch the old Bob Ross videos too. This is still all totally over my head though sadly. But for the first time ever, I feel like I could learn maths from someone. You seem right and the Bob Ross link is an ethereal helper. I'm going to look through your videos and see if there's a good entry point..... Thank you for doing these. I love that someone out there thinks a bit like I do! I had to school myself alone for reasons too detailed to go into but I still do the same thing. I take my ignorance and use it as a measuring tool to find the correct shape and size of a subject....
I don't understand a word, but the music of your voice is magic. So is the kindness and total beauty of U. Thank you Tibee, Soulmate of Bob Ross.
I had a Math teacher in College that scared me and my classmates during one year in College.
When i was playing during break with my friends, when the ring bell to stop the break we we're all scared in front of our classroom door.
Everybody was happy to go in class, except us. And i hated math because of her.
Gladly i did economy and sociology with help of other teachers and now i'm a teacher too in economy and sociology but it was difficult (because of maths and using statistics).
What i'm saying is... Your video prove Math can be teaching in a soft, calm and cheesy atmosphere. I think we could learn to do Maths during spring or summer , outside, around natural environment like you did.
It's amazing, congrats about your video.
Such a peaceful and soothing video.
It's actually funny though.
I'm impressed. That wasn't scary at all. You actually make mathematics look like fun.
This video is proof that even mathematics can be used to heal stress, not just music. By the way, when you said don't be scared by the equations on the board, i was greatly, " relieved " 😊😊😂😂😂
Its all Narcissitically so naferiously stressful!lol
I really appreciate the fairytale-esque charm of how you treat imaginary numbers. Thanks for a lovely video ✨
The proof is elegant and simple once the series expansion is taken for granted. Please do videos for series expansion proofs. Thanks so much. Never understood it my college days, its really simple , elegant and clear explanation.
Just brilliant!! :D as a physics student who likes to draw.. This is just priceless
i actually start to smile after watching this video. It stayed for rest of the day!! And you made me comment!!
Please Please do a video about how careers and study methods differ between a physict and engineer
Thanks
Your voice is so soothing! I hope it nets lots of people into learning about Euler's [wonderful] Formula!
For the first time in this life I could see that e^(i*pi) = -1 has three distinct constants all related by this equation. Thank you for making a fantastic video. I really like the scenic backdrop. It uplifts the mood immediately. I saw a Bob Ross video as well. That was also a treat.
I never miss your videos...
"Hold on to your calculators ladies and genteleman" 4:00
Magnificent !!! The same demeanor etc.
That smile - talk about melt your heart! And then the joyful wisdom from her lips, more treasure.
Actually we arn’t going to need many normal math teachers.. we will just have these famous online ones that are permanent features for humanity. They make learning their lovely subjects so fun & alive. The future humans will be so smart!!! Yay
Here's the perfect equation : (beauty + intelligence = Tibees) I hear so much of your interest in psychology that you mentioned in a different video coming through in the way you explain things. I love that.🙌
consider doing ASMR proving Mathematical equations ... that will put me to sleep instantly :)
I love all of your videos
i knew it! the female version of senpai is sin(π)
No. No. Just no.
Lol
hahaha
So senpai has no value?
I've been trying to understand this equation for years. It's so clear this way, thank you!
The taylor series, sequences, and recursive formulas are a thing of beauty. Whoever came up with this algebra to generalize an infinite sum into a general formula with finite concepts and to see if it converges to a finite value or infinite value is just marvelous. I especially like how you can add k+1 to prove your results as well. What is your favorite equation?