I am going to tear up in happiness as I have never been good with condensing notes to make things less overwhelming. That notes PDF is a game-changer for me. Thank you so much BPRP. I am so happy to have been subscribed to you all these years as you motivated me to become a math tutor. My goal is to make it up to Calc II and this will give me indescribable amount of value!
@@Archius_09 that man has no mate😂😂. Five of his kind does not exist on school work. His last video made understood MOMENT OF INNERTIA that I have being struggling with since 2019😂😂😂😂. BlackPenRedpen, You are Legit with Math Also. But that father is un measurable 😢
I enjoy watching your video. It's not just because you are good at explaining things but because we can see it in your face that you really enjoy making this video
4:38 Question 1 9:07 Question 2 14:00 Question 3 23:40 Question 4 30:20 Question 5 38:41 Question 6 44:40 Question 7 49:37 Question 8 53:10 Question 9 59:18 Question 10 1:05:25 Question 11 1:10:04 Question 12 1:16:03 Question 13 1:21:03 Question 14 1:28:43 Question 15 1:30:35 Bonus Question Guys the Check pdf in the description for the question. Give a like if you liked the hard work ❤️
My professor allows us to bring a notecard to the test and I wrote the series convergence tests table you provided on mine. I'm feeling pretty confident, thanks!
Hey! I just wanted to say thank you so much for this stream, my exam is in about 3 hours and I found this gem last night and binged through it. I feel so much more confident now!
watching this in 2023 a day before my calc 2 exam. thank you so much for your videos!! they really help me. btw i'm from nazarbayev university, kazakhstan
On the bonus question at the end, a bit of analysis can also be used. You can use the comparison test with the sum of 1/(2*n^2), which means the series converges to a value less than or equal to pi^2/12. The reason for this is because you can use the series expansion of 1 - cos(1/n) to get the sum of 1 - (1 - 1/(2*n^2) + O(1/n^4)), which simplifies to the sum of 1/(2*n^2) - O(1/n^4). It's less than or equal to the sum of 1/(2*n^2).
Even easier: notice that 1-cos(1/n)=2sin²(1/(2n)). This converges by #9 on this video, which showed that sin²(1/n) converges by the limit comparison test with 1/n². Since sin²(1/n) is always positive and it converges, then it does so absolutely. Since it converges absolutely, then any rearrangement of its terms also converges absolutely, and to the same value. Therefore, the sequence of even terms of sin²(1/n), sin²(1/(2n)), must also converge. Or you could just do limit comparison with 1/n² again lol. You basically did the same thing anyway, but it's even easier if you just use the trig identity so you don't have to worry about the Taylor Polynomial approximations and such.
Students the _"scary sigma"_ ∑ ☠ takes a lot of practice to master and BPRP has done a marvelous job here putting this material together covering all the various tests you need to know. ✔ *BonusQ:* LCT: [sin(1/2n)]^2 ,which is monotonic↘, with lim n→∞ of p-conv (1/(2n)^2), let ⌀=1/(2n) so lim ⌀→0 [sin(⌀)/⌀]^2=1.
If the ratio test is inconclusive, you might be able to use Raabe's test (of course it's not certain that it'll be conclusive either). You could add it in the notes after the ratio test.
It basically says that if the ratio limit is equal to 1, calculate the lim of n(1-an+1/an),let's say that this limit is equal to L, then if L>1 then the original series converges,if L
For bonus question, can use the double angle formula to change it to 2sin²(1/2n). Last we recall, there is a similar question earlier where we use limit comparison test for 1/(4n²)
For the bonus, I used Taylor series. Cos u = 1-u^2/2. So 1-cos(1/n) goes to 1/2n^2. Thanks to the p-series, because p is equal to 2 in this one, the bonus serie converges.
I have a test on Sequences on Series due today (I'm taking Calc 2 online through my local community college since I'm still in high school) so I'm using this video as a final review. I already watched it through once a few months ago when I was teaching myself, but I'm back again!
Yo blackpenredpen! For your next 100 problem challenge, could you tackle trigonometric equations and identities and stuff? Basically, formulas like sinC + sinD, the half angle formulas, and all that good stuff used in simplifying and solving problems. Thank you for everything.
Bonus Question: 1-cos(1/n)=2sin^2(1/2n) multiply divide by 4n^2 take limit sinx/x=0 as x-->0 we will be left with 1/n^2 variable term which we can take as vn now un/vn=1/2 hence by P series Vn is convergent and so is Un! :)
Are you able to give some information about dynamic systems? matrices etc. I have this question but unsure what to put as X1, Y1 and X2,Y2 "David jogs either in the woods or on asphalt. If he jogs in the woods, the probability is 30% that the next jogging trip is also in the forest, while the probability is 70% for the next jogging trip on asphalt.If he jogged on asphalt, the probability is 50% that the next trip is in the woods, and 50% the first trip on asphalt. David jogs his first trip in the woods. Find a formula for probability vector k, which describes the probabilities of forest or asphalt on trip k."
Float vs Natural Number vs Integer Range divided or not by consant vs fractional sets vs OE set vs Complex numbers vs Phasors how fast/ops for convergence vs error cases 1/INF=Infintesimal 10bit 1/INF=0 32bit
Preparing for analysis 1 exam in IT For the number 12 I did a Limit comparison test with 1/n²+1, which converges because it's less than 1/n² which, again, converges because it's a p series where p>1. So from the LCT I got π/2>0, so the 12 converges too. The solving should be correct, so yay I found another way which, to be honest, seems way easier than getting the graph, taking the integral et cetera.
I have 2 questions. Im not studying maths yet (Im still in school) so I dont know if my ways of solving are correct. Would be really cool if somebody could verify or show the mistakes. Question 12 arctan(n)
Isnt the 3rd question going to be solved by Leibnitz test cause the de alembert he is using is only for the series were the terms are not of negetive signs
At 1:23:10 Could you also say that the term (1/n - 1/(n+2)) has to be smaller than 1/n =1/n^1 and therefore conv by P-Series since a theoretical p would have to be greater than 1 for the term to be smaller than 1/n. I hope you get what I mean
At least I think the criteria for the P-Series imply, that if the terms of a function that are added in the sum go faster towards 0, than they go when adding the terms from 1/n, the function you are looking at converges
Could question 14 1:27:30 , also be written as the series from n=2 to infinity of 1/(n^2-1), and use LCT to show convergence? (If you didn’t care about finding the sumn
Hey guys quick question: in question 3 he used the ratio test to determine the behavior but isn't the ratio test applicable only to positive series? i mean shouldn't he use the leibniz criterion for alternating series for that one?
I must be extremely dumb, but I still don't understand when and what types of series to apply to problems. As far as i can tell there is no general rule in knowing which one to use. It all has to be memorized??? Can someone please help me out, there has to be some pattern here i'm missing?
Get the file and the notes first 👉 www.patreon.com/posts/102348401
Sirr does it need for the note you posted a money
I am going to tear up in happiness as I have never been good with condensing notes to make things less overwhelming. That notes PDF is a game-changer for me. Thank you so much BPRP. I am so happy to have been subscribed to you all these years as you motivated me to become a math tutor. My goal is to make it up to Calc II and this will give me indescribable amount of value!
It looks interesting but I can't understand a word you say.
Can you share it with me please?
The video thumbnail reminded me of Organic Chemistry Tutor
at first I taught it was him but then I saw the thumbnail carefully and then channel name
OCT is literally god
Broo trueee
@@Archius_09 that man has no mate😂😂. Five of his kind does not exist on school work.
His last video made understood MOMENT OF INNERTIA that I have being struggling with since 2019😂😂😂😂.
BlackPenRedpen, You are Legit with Math Also. But that father is un measurable 😢
I got click baited😢
omg literally thank you SO much. I know it's been two years since you made this live but it came to my life in the perfect time. You're AMAZING !!!
I enjoy watching your video. It's not just because you are good at explaining things but because we can see it in your face that you really enjoy making this video
Thanks!
Timestamp
4:38 Question 1
9:07 Question 2
14:00 Question 3
23:40 Question 4
30:20 Question 5
38:41 Question 6
44:40 Question 7
49:37 Question 8
53:10 Question 9
59:18 Question 10
1:05:25 Question 11
1:10:04 Question 12
1:16:03 Question 13
1:21:03 Question 14
1:28:43 Question 15
1:30:35 Bonus Question
Check the pdf for the questions
Integrate. [Cos^-1x (√1-x^2)]^-1 / Log{1+(sin(2x√1-x^2)/π}
If you are interested in mathematics here is my videos on sliderule
m.th-cam.com/video/ReNFu4meONc/w-d-xo.html
almost
Sir from where I can download the pdf of those questions ?
In question 4, why did you replace sin(2n) by 1?
Thank you!
love how enthusiastic this man is about calculus
Can anyone help me with the timestamps? Thank you.
4:38 Question 1
9:07 Question 2
14:00 Question 3
23:40 Question 4
30:20 Question 5
38:41 Question 6
44:40 Question 7
49:37 Question 8
53:10 Question 9
59:18 Question 10
1:05:25 Question 11
1:10:04 Question 12
1:16:03 Question 13
1:21:03 Question 14
1:28:43 Question 15
1:30:35 Bonus Question
Guys the Check pdf in the description for the question. Give a like if you liked the hard work ❤️
@@treanungkurmal803 Thank you so much Treanungkur!
@@blackpenredpen Really pleased sir that you replied ❤️
My professor allows us to bring a notecard to the test and I wrote the series convergence tests table you provided on mine. I'm feeling pretty confident, thanks!
BPRP you are a blessing, I have a series test tomorrow and this is helping me a lot to practice!!! Keep up the great content!
This is EXACTLY what I needed. I’m struggling with this topic at the moment
Thanks
This video was incredible. Used it for my exam last week and it was a CRUCIAL piece of my success
Your good mood is so contagious! Thanks a lot for your clear explanations and giving people like me enjoyment doing this!!
44:22 thanks for the shout out my guy
Ey thas my friend
Number 14
Can also be expressed as sum from 2 to inf of 1/(n^2-1)
Thank you for your kindness by posting many videos for educational purposes. You are good man !
Someone who truly loves math, I found my polar opposite.
Hey! I just wanted to say thank you so much for this stream, my exam is in about 3 hours and I found this gem last night and binged through it. I feel so much more confident now!
How did you do
@@ioannischrysostomou7012 Yeah.... I failed
watching this in 2023 a day before my calc 2 exam. thank you so much for your videos!! they really help me.
btw i'm from nazarbayev university, kazakhstan
A truly fine beard my good sir!
On the bonus question at the end, a bit of analysis can also be used. You can use the comparison test with the sum of 1/(2*n^2), which means the series converges to a value less than or equal to pi^2/12. The reason for this is because you can use the series expansion of 1 - cos(1/n) to get the sum of 1 - (1 - 1/(2*n^2) + O(1/n^4)), which simplifies to the sum of 1/(2*n^2) - O(1/n^4). It's less than or equal to the sum of 1/(2*n^2).
Even easier: notice that 1-cos(1/n)=2sin²(1/(2n)). This converges by #9 on this video, which showed that sin²(1/n) converges by the limit comparison test with 1/n². Since sin²(1/n) is always positive and it converges, then it does so absolutely. Since it converges absolutely, then any rearrangement of its terms also converges absolutely, and to the same value. Therefore, the sequence of even terms of sin²(1/n), sin²(1/(2n)), must also converge. Or you could just do limit comparison with 1/n² again lol. You basically did the same thing anyway, but it's even easier if you just use the trig identity so you don't have to worry about the Taylor Polynomial approximations and such.
thank you balckpenredpen.............
for posting this days after my exam....
but seriously good video
Students the _"scary sigma"_ ∑ ☠ takes a lot of practice to master and BPRP has done a marvelous job here putting this material together covering all the various tests you need to know. ✔
*BonusQ:* LCT: [sin(1/2n)]^2 ,which is monotonic↘, with lim n→∞ of p-conv (1/(2n)^2), let ⌀=1/(2n) so lim ⌀→0 [sin(⌀)/⌀]^2=1.
Super proud of myself of going through this start to finish
THANK YOU! for that gift note!
BLESS! These notes are going to make my homework downright bearable!
In question no 3 14:00 -2/e is correct as minus sign will remain with 2
no, because of the abs value
Thanks for this, test over it tomorrow!
For number 3, the root test would work as well, and includes a famous limit (n!)^(1/n) / n as n approaches infinity.
13 Question , i think it is conditionally Convergent as |Un|=1/n(3+1/n) where by p series p=1
If the ratio test is inconclusive, you might be able to use Raabe's test (of course it's not certain that it'll be conclusive either). You could add it in the notes after the ratio test.
It basically says that if the ratio limit is equal to 1, calculate the lim of n(1-an+1/an),let's say that this limit is equal to L, then if L>1 then the original series converges,if L
your amazing!!! you make this really easy to understand and follow along !!
Saved my calc 3 test ❤
You are the legend, always the best explanation
Great job man
For bonus question, can use the double angle formula to change it to 2sin²(1/2n). Last we recall, there is a similar question earlier where we use limit comparison test for 1/(4n²)
WHY WAS THIS NOT RELEASED 4 DAYS AGO WHEN I DIDNT HAVE MY CALC 2 EXAM ON FRIDAY
-purdue engineering student
Haha I really could have used this practice before I failed my calc final yesterday :)
I have mine today luckily
For the bonus, I used Taylor series. Cos u = 1-u^2/2. So 1-cos(1/n) goes to 1/2n^2. Thanks to the p-series, because p is equal to 2 in this one, the bonus serie converges.
Cos(1/n) at n tends to inf =1 hence divergent
You really motivate me and I wish to meet you in person someday 😊
Integrate. [Cos^-1x (√1-x^2)]^-1 / Log{1+(sin(2x√1-x^2)/π}
Bro if I pass my first analysis test in uni it's all because of you. Thank you so much for the content!
I'm currently doing real analysis and this video was soo helpfull thanks so much
YOU ARE AMAZING!! You are saving my calc II grade in university.
i have my calc 2 final exam in 4 days bro actually saved me
I have a test on Sequences on Series due today (I'm taking Calc 2 online through my local community college since I'm still in high school) so I'm using this video as a final review. I already watched it through once a few months ago when I was teaching myself, but I'm back again!
Yo blackpenredpen! For your next 100 problem challenge, could you tackle trigonometric equations and identities and stuff?
Basically, formulas like sinC + sinD, the half angle formulas, and all that good stuff used in simplifying and solving problems.
Thank you for everything.
That's on my list! : )
Abbreviation may change between teachers for the final test explanation (problem 4), but this is very helpful! Thank you bprp
This was just wonderful :D thanks a lot
From Lebanon 🇱🇧
Thank you very much❤️
Heyy there i'm too from Lebanon!
Are you in uni?
@@elie.makdissi yes
@@elie.makdissi first year
Engineering
At USEK
@@mirayayna1295 Ah cool gd luck!
Hi me to arabic but I'm from morocoo
Bonus Question:
1-cos(1/n)=2sin^2(1/2n) multiply divide by 4n^2 take limit sinx/x=0 as x-->0 we will be left with 1/n^2 variable term which we can take as vn now un/vn=1/2 hence by P series Vn is convergent and so is Un! :)
Ratio test is called d’alembert rule in 🇫🇷
For 12, you can just do direct comparison test with pi/(2x^2) which converges as arctanx is bound by pi/2.
Thank you from Italy!
How to limit X tends to zero sin(X) also tends to zero. In problem no 9
Your pdf is very very helpful to me sir thank you for your effort for us.
I have my final exam tomorrow. I’m channeling my bprp knowledge before bed with this video
This is awesome thanks professor!!!!!!!
Another great video!!
Thank you so much you saved my life!
Happy to help!
Are you able to give some information about dynamic systems? matrices etc.
I have this question but unsure what to put as X1, Y1 and X2,Y2
"David jogs either in the woods or on asphalt. If he jogs in the woods, the probability is 30% that the next jogging trip is also in the forest, while the probability is 70% for the next jogging trip on asphalt.If he jogged on asphalt, the probability is 50% that the next trip is in the woods, and 50% the first trip on asphalt. David jogs his first trip in the woods.
Find a formula for probability vector k, which describes the probabilities of forest or asphalt on trip k."
5 can also be done by integral test
Bonus question answer. Converges by DCT compare it to 1/(n^2) (a convergent series) which is bigger than 1 - cos(1/n).
Ps: thank you so much for this video!
bonus >> LCT with 1/n^2 , it will converge
No. It will diverge by LCT taking 1/2n
@@riazuddinsheikh3198 You mean diverge by using LCT, and comparing by 1/2n
your videos are amazing. Thank you.
Float vs Natural Number vs Integer Range divided or not by consant vs fractional sets vs OE set vs Complex numbers vs Phasors how fast/ops for convergence vs error cases 1/INF=Infintesimal 10bit 1/INF=0 32bit
me watching this after getting 50% of them on my homework 🥴
Life saver! Thank you so much.
Wow I have an exam of series tomorrow thanks😂😂😂❤
Preparing for analysis 1 exam in IT
For the number 12 I did a Limit comparison test with 1/n²+1, which converges because it's less than 1/n² which, again, converges because it's a p series where p>1. So from the LCT I got π/2>0, so the 12 converges too. The solving should be correct, so yay I found another way which, to be honest, seems way easier than getting the graph, taking the integral et cetera.
Thank you 🌈👌
Q bonus : use Taylor series.
1 - cos (1/n) = 1 - 1 + 1/n^2 + o(1/n^4)
Equivalent to 1/n^2 serie which converges.
Damn you make me miss math
Loov maht
I have 2 questions. Im not studying maths yet (Im still in school) so I dont know if my ways of solving are correct. Would be really cool if somebody could verify or show the mistakes.
Question 12
arctan(n)
Your method for #12 is close but not correct. Notice that arctan(1)=π/4 and arctanx approaches π/2 as x→∞, so arctan(n)
26:00 why didnt we do 1/n > 1/(n+3^n)
This is amazing thank you
very helpfull video thanks Mr. bprp
Thank you! You're the best!
So nice teaching bro .. thank you so much .... 👍🏼
This guys dope. I like him!
Thank you so much!!!
Thank you so much!
This is good practice for my final!
n4+3n/4n3-2n+1 divergence or convergence
I thought that if cos(0) is 1 and cos(1) is about 1/2 you would be adding closer and closer to one and the sum would be divergent.
Only if I had u as my maths teacher during my grad.
maths forever ❤️❤️❤️
Isnt the 3rd question going to be solved by Leibnitz test cause the de alembert he is using is only for the series were the terms are not of negetive signs
I come here for the T-shirts. Best on TH-cam. 🤓🤪
What could make BPRP better ? Subtitles for SIX languages .
Math Fhein a Thidseir .
Well done Teacher .
At 1:23:10
Could you also say that the term (1/n - 1/(n+2)) has to be smaller than 1/n =1/n^1 and therefore conv by P-Series since a theoretical p would have to be greater than 1 for the term to be smaller than 1/n.
I hope you get what I mean
At least I think the criteria for the P-Series imply, that if the terms of a function that are added in the sum go faster towards 0, than they go when adding the terms from 1/n, the function you are looking at converges
I had to learn this in a week over summer 😩 damn now I forgot it all
loving the beard!
Thank you very much.clear and sweet.
Could question 14 1:27:30 , also be written as the series from n=2 to infinity of 1/(n^2-1), and use LCT to show convergence? (If you didn’t care about finding the sumn
Thank you :D
I got a doubt for question 2: shouldnt 'a' suppose to be 1 instead of 7/8 . so I end up with 1/ (1-7/8) = 8 . so the series converges to 8
Nice beared sir
Hey guys quick question:
in question 3 he used the ratio test to determine the behavior but isn't the ratio
test applicable only to positive series? i mean shouldn't he use the leibniz criterion for alternating series for that one?
No my friend, the ratio test (if conclusive) determines either absolute convegence or divergence.
I love this video
I must be extremely dumb, but I still don't understand when and what types of series to apply to problems. As far as i can tell there is no general rule in knowing which one to use. It all has to be memorized??? Can someone please help me out, there has to be some pattern here i'm missing?