I know that this is convergent. I've proven it with mathematical induction but I want to know what is the exact limit.I have no idea what to do. can anybody help me? Lim $1+\frac{1}{4}+\frac{1}{9}\ldots \frac{1}{n^2}$
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Here's a MathJax tutorial :) – Shaun Sep 06 '17 at 18:41
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2see here https://en.wikipedia.org/wiki/Basel_problem – Dr. Sonnhard Graubner Sep 06 '17 at 18:41
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1The answer: $$\frac{\pi^2}{6}$$ – MCCCS Sep 06 '17 at 18:41
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1https://en.wikipedia.org/wiki/Basel_problem – Angina Seng Sep 06 '17 at 18:41
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It isn't enough to speak of "the limit": you need to specify where $n$ is heading (in this case, "as $n$ is approaching infinity"). So instead of $\lim 1+\frac14+\cdots+\frac1{n^2}$, write $$\lim_{n\to\infty} 1+\frac14+\cdots+\frac1{n^2}.$$ – Théophile Sep 06 '17 at 18:53
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See wikipedia for a myriad of solutions to the problem:
https://en.wikipedia.org/wiki/Basel_problem
You will see that they require specialized knowledge in some form or another.
Also, this paper which goes into detail about several different proofs:
https://www.uam.es/personal_pdi/ciencias/cillerue/Curso/zeta2.pdf
But, because you'd like to know the limit
$$\sum_{k=1}^\infty \frac{1}{k^2} = \dfrac{\pi^2}{6}$$
Chickenmancer
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