I’ve recently found out that $\sum_{n=1}^{\infty}\frac{1}{n^2+n}$ makes 1, since it becomes $\frac{1}{2}, \frac{2}{3}$ and so on. After then, I’ve became curious if I do the same thing with the reciprocal of n squared, or $$\sum_{n=1}^{\infty} \frac{1}{n^2}$$ I couldn’t find out the answer. Their sums don’t make a neat form like $\frac{a}{a+1}$. Could anyone tell me what it approaches to?
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3This is a pretty well-known problem! Check it out here: https://en.wikipedia.org/wiki/Basel_problem – boink Jul 28 '20 at 01:06
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1Extremely impressive that your'e familiar with summations and sigma notation at your age. Keep it up! :) – A-Level Student Jul 28 '20 at 16:08
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The sum is $\pi^2/6$. Euler first figured that out. It's no surprise and no disgrace that you didn't. See https://en.wikipedia.org/wiki/Basel_problem .
Ethan Bolker
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Sorry, but I’m just a 10 year old boy who’s just interested in maths and I can’t understand those complex theories and symbols in wikipedia. – Jul 28 '20 at 01:11
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6It's a very nontrivial problem, Byun. Don't push yourself; you'll understand it one day. For what it's worth, 3Blue1Brown motivated a nice, geometric proof of the sum here -- https://www.youtube.com/watch?v=d-o3eB9sfls – PrincessEev Jul 28 '20 at 01:13