Calculus 2 - Integral Test For Convergence and Divergence of Series
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- เผยแพร่เมื่อ 28 ก.ย. 2024
- This calculus 2 video tutorial provides a basic introduction into the integral test for convergence and divergence of a series with improper integrals. To perform the integral test, let the sequence be equal to a function that is continuous, positive, and decreasing on the interval [1, infinity). If these conditions are met, then the infinite series will converge if the definite integral from 1 to infinity converges. If the integral test is divergent, then the series is divergent as well. Examples and practice problems include integration techniques such as u-substitution, power rule for integration, trigonometric substitution which leads to inverse tangent functions, and completing the square before integration. The divergent harmonic series is included in this video as well.
Improper Integrals:
• Improper Integrals - C...
Converging & Diverging Sequences:
• Converging and Divergi...
Monotonic & Bounded Sequences:
• Monotonic Sequences an...
Absolute Value Theorem - Sequences:
• Absolute Value Theorem...
Squeeze Theorem - Sequences:
• Squeeze Theorem For Se...
________________________________
Geometric Series & Sequences:
• Geometric Series and G...
Introduction to Series - Convergence:
• Convergence and Diverg...
Divergence Test For Series:
• Divergence Test For Se...
Harmonic Series:
• Harmonic Series
Telescoping Series:
• Telescoping Series
__________________________________
Integral Test For Divergence:
• Calculus 2 - Integral ...
Remainder Estimate - Integral Test:
• Remainder Estimate For...
P-Series:
• P-series
Direct Comparison Test:
• Direct Comparison Test...
Limit Comparison Test:
• Limit Comparison Test
___________________________________
Calculus Final Exam and Video Playlists:
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Full-Length Videos and Worksheets:
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i appreciate that he doesn't skip steps because he thinks its "easy" or everyone should know how to do that step. I have a lot of smart professors who think people are as smart as them in a subject and fail to go over all the steps, which makes it harder for everyone to keep up
My professor is like this and I think that's the main reason I don't understand his classes while I am perfectly fine over here, some student had the courage to ask how did he get that result because he skipped like 3 steps and he ridiculed the student in front of everybody for not understanding the steps he skipped because we should have learnt all of this perfectly in highschool...
I know that feel mate...
@@akin.kilic. damn
Those are the dumb professors in my opinion. The smart ones know that the students aren't as good at math as themselves and will work out every step
My professor while a smart guy and a good teacher, he will constantly go over steps and it sometimes takes me a couple moments to figure out what shortcut he took. While I'm doing that he is already moving on with the material and it can be annoying.
This man has single handedly saved my ass in multiple subjects. Mad respect my guy, you're getting me through calculus.
He’s got this “Dora the Explorer” pace where he wants you to answer yourself lol
im glad that there is a speed option, 1.25 speed rules for (almost) all learning tutorials
@@tv-fz6ws we get it
For the last problem, it was supposed to be π-2arctan(4).
correct, because of the factor of 2 infront of the arctan(inf)
God bless both, you and Chegg.
Chegg 2greedy
@@nonyabizness577 chegg teams up with the people who sell you the questions so they can sell you the answers, haha. Protest them
Chegg.... yea right. I use Slader.
Chegg sucks. Just cancel their membership and they will give it to you for $4.99 a month. Rinse and repeat.
Don’t forget about the mans Sal from Khan Academy
should change your name to "the everything tutor"
27:48 shouldn't the 2 in front cancel out the 2 from pi/2? Final answer should therefore be pi-2 arctan(4)?
Yeah I noted that too, I don't know I think he mis that step
Yea I was just thinking that
I just checked with an integral calculator and he made a mistake, the final answer should be [π-2tan⁻¹(4)]
You are true 👍❤
brooo I hope you pass your classes.
This is real art of Mathematics
best lesson ever i'm giong to ace my upcoming tests
did you ace it?
"the organic chemistry tutor"
who apparently knows everything. I love this guy hahahahahaha
there is so much information to absorb in calculus. he always goes over simple things that i have already forgotten ....
He absolutely explains everything
For the last example, is the final answer not PI minus 2 times arctangent of 4? Since you have the constant 2 in front of the whole thing
For the series 2n/3n^2 + 4:
Why do the first derivative test to check if it's decreasing? Wouldn't you just be able to check if it was decreasing by seeing that the power on the bottom is greater than the power on the top? Or, if you just plugged in the first and second terms. s1 = 2/7 > s2 = 1/4, therefore its decreasing.
If the integral from 1 → infinity of 2x/3x^2+4 was evaluated, you could do u-sub and you'd get the same answer but with way less work.
It depends on your teacher and how pedantic they are. Since he doesnt know who your teacher is he teaches the long route so everyone is covered.
@@Epicvampire800 Oh! Nice. I was wondering the same thing
Nice video
Thank you sir, you are an inspiration ✨
@13:43, how do you know that it is continous at [2, infinity)?
for every input value on interval [1, inf ) there is always an output
It's high time you changed your TH-cam name to MegaMind sir! 🧠🧠
I respect this channel
I am still confused about replace the (x+3)^2 with tan theta
This is wonderful
Great video! Thank you so much for making it. God bless :)
at 27:54 did the tutor for got to put 2 on pi/2? because it has 2 in than^-1 (b+3)
Barakalla 🤲 jazakkalla
A lot of these we don't even need to test if they're decreasing since bottom heavy obviously means decreasing
why is not necesssary to find the limit of convergence of the series? we prove with the test that there is a converg. or divergence. how to find the convergence of the series?
Thanks as always 😊
after 6 minutes i finally heard the "you can now use the integral test" kil me -_-
i am so gon fail
If you even remember after 4 years, how did it go? 😂
Are you alive???
we need to know @mosubaldakheel9245
uh question: Didn't the 2's cancel each other out at the end?
Thank you for going into detail btw!
I guess they did, was looking for a comment if anyone else has realized
at 24:00 why did you not just use u sub for x+3 instead of trig sub, is it because how profs teach it or something because 1/u+1 du is inverse trig so it would be 2tan^-1(u) and u is x+3
oh thank god arctan is in here
Excellent
THANK YOU SO MUCHHHHHH
theres more effeicient ways to tell if the function is decreasing
12:02 Is he saying "sine chart"?
(That's what the auto-generated subtitles say)
sign, not sine.
sorry mr. 22:00 Minuit example that u did is not correct if u can try to edit that one
I LOVE YOU!!!!
27:49 I guess 2tan(inf) should be pi instead of pi/2
yes ur right, i was thinking the same thing too!
thannks!
why did he choose tanθ to replace x+3??? I know he said the other form related to arctan but why did he choose that specifically??
you could also u sub: u = x+3 and find that integral 2/u^2 + 1 which is in form of antiderivative of 1/u^2 + 1 = tan^-1x so 2tan^-1(x+3)
saved once again..
when you do u-sub, dont you have to make your new points on the definite integral?
Optional
Last question, it's π you forgot to multiply the 2
Could someone explain why we differentiate 2x/(3x^2 + 4), instead of just integrating with the given function and solving for it at infinity and 1? Basically, why is the second problem solved differently than the previous one?
Thank you!
We differentiated so we could use the first derivative test to see if the function decreases on [1,infinity)
Why does it feel like this video was made in 2007
Sadly, there are too many ads. It disrupts the learning process, and the ads constantly intrude at inconvenient times. If you could please reduce the number of ads, or limit them to the beginning of the video and only at one point in the middle, that would work best. The ads keep popping up every 5-8 minutes, which is quite frustrating.
This guy is amazing. He mastered almost every subject.
I have the same idea
Is the avatar of science
If he teaches one more discrete math that will be perfect
This guy should change the name to EVERYTHING tutor
This mans channel has single handedly gotten me through calculus 1 & 2. I love you organic chemistry tutor! I wish good fortune for you and your loved ones
is there any way i can donate to you? I would love to, youre literally carrying me through calc and physics right now
He's got a Patreon linked in the description
lets help him on his other platforms and also by not skipping ads
no time for ads when exams coming up srry
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plz upload more videos on multivariable calculus. you helped me pass calc 2
I do not know how I would be passing without your videos. Thank you so much!
dude u have no idea how many times u saved my grades from c and below not just in school but in college too its because of u that entered science school and I even joined the math department 😁😁
Professor Organic Chemistry Tutor, thank you for another fantastic video/lecture on the Integral Test for Convergence and Divergence of Series in Calculus Two. This part of Calculus Two can be problematic, however students must do many problems in order to fully understand this material. This is an error free video/lecture on TH-cam TV with the Organic Chemistry Tutor.
23:39 if u dont understand the trig sub:
if a function looks like (1/[(a^2)+(u^2)])
where a^2 is some constant term and u is some variable
arctan formula takes the form 1/a * arctan(u/a)
for this problem.
take the two in the numerator out of the integral
replace (x+3) with u
a = 1
2 * (1/1 * arctan[(x+3)/1])
12:20 Don't waste time "testing" the regions. Just quickly graph (since the denominator is always positive) y=8-6x^2 which is easy since u have the roots (and u understand end behavior) and you'll get a parabola that opens down.
Most college calculus courses don’t allow you to use a calculator for quizzes and exams. This is a hand drawn method for those who can’t use a calculator or understand the methods behind it. But I totally get your point
You're my favorite person
Bro you saved me the past 9 years😭😪..I'm having my final exams next week
If you can graph the function and can see that it decreases will that suffice to show that the function decreases or must you take the first derivative?
To the mathematician who invented integral test , I hope you're good in he*ll , bro improper integrals is enough why would you do this again ?
Thanks a lot.
From DRC,i have enjoyed it.
I hope I can someday return to this video and say "I really had to watch this to pass calc 2, now all of my classes are just coding ones.."
What should we do if the denominator is non factorisable quadratic
I believe that you could have just done a u-sub instead of a trig sub for the last problem, would have given you 1/(u^2 + 1) which is the derivative of arctan, plug the u back in for your integral. Great video though.
Arctan is like the definition of a trig sub…
You are amazing! Thank you so much!!!!!!
excellent explanation as usual
mathematical induction
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its mindblowing that a guy on the internet manages to explain things better and quicker than the "top professors" in universities...
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the last limit is supposed to be π-2acrtan(4)
Love how you make things simple
How did du comed 6x ???
Finals in 3 hours
Please help Summation ln(n/n+1)) determine if it converges
Please how do you get to know the genre of test you're to go about or use if the question doesn't state it ?
Why tf would we use derivative when it's literally integral test 12:49!??
super epic resource thank you v kindly
Simply the best teacher out there
I have a question if the integral converges to an N value does the series also converge to that N value or do we just know that it converges ?
They don’t converge to the same value
Why are you not changing the limits of integration after making the u-substitution?
as long as you sub in for u at the end, you're changing the integral back to terms of x, so you can use the same bounds for x when evaluating that integral
13:29 could you / anyone explain why we have to now focus from 2 to infinite
We have a condition that our function has to be decreasing on the intervals [1, infinity). But we saw that it is not. Instead of this we took from 2 to infinity because it is decreasing in this interval. (im not sure about it 100%, and sorry for my english)
lov u
Someone tell me if I'm wrong but can't you easily tell if a fractional function is decreasing by simply looking at the exponents on top and bottom and seeing if the order of the bottom is larger than the top? I know why he is using the first derivative test to do the proper solution but I'm sitting here pulling my hair out wondering why he hasn't at least mentioned the shortcut in passing.
Dont forget calculator
Is this really calc 2? I am taking calc 3 right now and just learning this
yes. im taking cal 2 rn and im learning this.
@@logicbewithyou111 oof I have a calc 3 test on this lol but after I this I think I will learn power series
In the second example, why does du becomes 6x dx? Should it be just 2x dx?
pls do we know why he chose to replace x with b, just curious. It seems that x will still be fine to me
he is using improper integration and b refers to the larger number which here is infinity
nice explanation
What if we get a negative 0? Is it convergence or divergence? Hope you'll see this. Thank you.
Negative 0 is 0.
are u dumb
Any good explanation for LIZARD'S
if the resulting integral value is zero in an integral test, does the series converges?
MI MUM MADE MEWATCH DIS WHEN I WAS 15 NOW AM 17 FORCED BY MY PROES TO WATCH DIS
Sir what if you take the interval of the 2nd Question from 1 to infinity
What happen if n starts from 2
Does it satisfy the integral test
The final answer you forgot to multiply 2 in pi/2
Am I the only one that feels his voice is so satisfying ?
extreme praise for you!!
any one know about the integral test applications