# Linked Questions

2answers
37k views

### Showing $\sum _{k=1} 1/k^2 = \pi^2/6$ [duplicate]

Possible Duplicate: Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$ Does $\sum\limits_{k=1}^n 1 / k ^ 2$ converge when $n\rightarrow\infty$? I read my book of EDP, and ...
4answers
5k views

### Complex Analysis Solution to the Basel Problem ($\sum_{k=1}^\infty \frac{1}{k^2}$) [duplicate]

Most of us are aware of the famous "Basel Problem": $$\sum_{k=1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}$$ I remember reading an elegant proof for this using complex numbers to help find the value of ...
1answer
28k views

### Sum of $\frac{1}{n^{2}}$ for $n = 1 ,2 ,3, …$? [duplicate]

Possible Duplicate: Different methods to compute $\sum_{n=1}^\infty \frac{1}{n^2}$. I just got the "New and Revised" edition of "Mathematics: The New Golden Age", by Keith Devlin. On p. 64 it ...
1answer
5k views

0answers
338 views

### Proving $\int_0^1\frac{\log (1-x)}{x}\mathrm dx=-\frac{\pi^2}6$ [duplicate]

It is a well known fact that $\displaystyle\sum_{k=1}^{\infty}\frac1{k^2}=\frac{\pi^2}6$. I wanted to prove this using elementary techniques. By doing some easy algebra, I found it was sufficient to ...
1answer
181 views

### How to calculate $S_{n}=\sum_{i=1}^{n}\frac{1}{i^{2}}$ [duplicate]

In math lesson, my teacher told me that Euler once used a very delicate method to calculate $\displaystyle S_{n}=\sum_{i=1}^{n}\frac{1}{i^{2}}$ and wrote a paper about it. I wonder how to calculate ...
1answer
198 views

### Show that $\sum_{n=1}^{\infty}{\frac{1}{n^2}}=\frac{\pi^2}{6}$ [duplicate]

Show that $$\sum_{n=1}^{\infty}{\frac{1}{n^2}}=\frac{\pi^2}{6}$$ Anyone can help ?
0answers
158 views

### using only power series to show $\sum\frac{1}{n^2}=\frac{\pi^2}{6}$ [duplicate]

Possible Duplicate: Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$ What is the simplest way to show $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$ using only power ...
3answers
111 views

I know that the series b. converges as $\sum \frac{1}{n^p}$ converges for $p>1$, So a. also converges. I want to know the sum. a.$1+\frac{1}{9}+\frac{1}{25}+\frac{1}{49}+.....$ $b.... 0answers 142 views ### Fault in proof of$\zeta(2) = \frac{\pi^2}{6}$[duplicate] Consider the proof of: $$\zeta(2) = \frac{\pi^{2}}{6}$$ So the proof assume that (because of Euler decomposition) $$\frac{\sin(x)}{x} = \prod_{n > 0}\left(1 - \frac{x^{2}}{(n\pi)^{2}}\right)$$ ... 0answers 119 views ### Is there an interpretation for this classic identity? [duplicate] Possible Duplicate: Different methods to compute$\sum_{n=1}^\infty \frac{1}{n^2}$.$\sum \frac{1}{n^2}= \frac{\pi^2}{6}\$ There are a few proofs for that fact but can anybody see why is it "...

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