Does sum 1/n^2 converge? - Week 2 - Lecture 11 - Sequences and Series

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I'm really lucky to have found these on Coursera. Your videos are always really thought-provoking and extremely interesting. The key thing is that you don't just spit out the formulas and the theorems but that you actually go through examples and proofs to show why things work the way they are. Great videos!

UWBadgersfan

Absolutely magnificent! Really informative and detailed and still compact and short explanations. Keep up the good work.

JohnnyBackwater

Nice work! I would have been tempted with: ∫ 1/x² < Σ 1/n² < ∫ 1/(x-1)² which can be evaluated from 2 to ∞, yielding: 3/2 < Σ 1/n² < 2 (for 1 to ∞), showing the sum converges. But your method avoids any integrals. Showing the result = π²/6 (Euler), however, requires more than comparison to numerical evaluation (surprisingly challenging...)

djconnel

we may also solve it by monotone convergence theorem.As 1/n^2 is monotonically decreasing and bounded by 2 either if {x_n}=1/n^s for all n belongs to N and as its decreasing and I X_n I <= 2 so this sequence is convergent.

rajarshichatterjee

Thanks a lot Jim; this video cleared it up a lot for me.

Your enthusiasm for the pi^2/6 answer was very thought provoking! These type of questions make math interesting for me and keep me motivated to continue to "learn the language". Thanks!

RobSummers

Hey
Can you give a proof of for n=1 to infinity ∑1/(n^2) = (pi^2)/6
I appreciate your work, Sir :)

piyushgone

If 1/n^2 < 1/(n^2 - n) then it means each term of the left hand side is smaller than each term on the right hand for the corresponding n value .. then how does the summation of the left hand be greater than the summation of the right hand ? 1.6 > 1 right!
Please clear my perception .. where am i missing out the understanding ?

suwathijain

niceee video...but m still confused about 1 thing..regarding pi^2/6
pi^2/6 =1.6449 (approx.) but how to find whether the series add up to this same no. only...what i mean to ask is like u showed the summation of the series till n=1000..which was 1.6439...but considering n tends to inf. how can we really terminate the sum till 1.6449. I can see that the sum won't exceed 2 (it converges at that point) but can't the sum equal 1.7 or even 1.8 n something like that...'why 1.6449..how do we find that?' is my question. Hoping for a helpful response soon.

twinklerathi

wow that was a huge help. Thank you so much. Now I can finish my assignment

milessage

Are you going to make a video about the strange thing with pi^2 / 6? If so, when?
Love your work by the way you're a big help for me as I'm preparing for my first exam in analysis

ggPescesgg

I remember my calc 2 teacher telling us this and it blew my mind.

RichardJohnson_dydx

thanks, i was thinking about the integral from 1 to infinity of 1/x^2 as a possible proof that the series converges. could i use this argumentation?

xavierpoes

You look like my Group theory lecturer... and well done

dapulaburger

How would you go about proving this using the definition of a limit?

aronoco

But how come when I take the improper integral I get positive 1?

Coolgiy

how can you hold your pen like that? I've been trying to replicate that grip but my fingers literally can't do it.

kevinmckay

How can 1/n^2 equal to 0 0:59 in second line

sakshamrana

how do you know that the sum of 1/(2)^n is 2?

goodbackgroundmusic

sequence =2/nsquare is a cauchy . how to prove

chaudharyprabhat

Please change the paper!!! The sound It bothers me...

julioardenghijr

Does sum 1/n^2 converge? - Week 2 - Lecture 11 - Sequences and Series

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