Linked Questions

9
votes
4answers
6k 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 ...
6
votes
2answers
43k 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 ...
1
vote
1answer
38k 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 ...
2
votes
1answer
10k views

Why does the sum of inverse squares equal $\pi^2/6$? [duplicate]

I've heard that $1+1/4+1/9+1/16+1/25+...$ converges to $\pi^2/6$. This was very surprising to me and I was wondering if there was a reason that it converges to this number? I am also confused why $\...
0
votes
1answer
2k views

What is the sum of the series $\sum\limits _{k=1}^\infty \frac{1}{k^2}$? [duplicate]

Possible Duplicate: Different methods to compute $\sum_{n=1}^\infty \frac{1}{n^2}$. What is $\lim \limits_{n\to\infty} \sum\limits_{k=1}^n \frac{1}{k^2}$ as an exact value?
2
votes
3answers
343 views

How to prove that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ without using Fourier Series [duplicate]

Can we prove that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ without using Fourier series?
1
vote
2answers
301 views

Proof for $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$ without complexes? [duplicate]

This is what I needed. Practically, a link were also okay. $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$
0
votes
1answer
248 views

Prove that $\sum \frac{1}{n^2} = \frac{\pi^2}{6}$ [duplicate]

In this answer two sequences are mentioned. In particular, I would like to prove that $$\sum_{n = 1}^{+ \infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$ If I knew that the sequence converges to $\frac{\pi^...
8
votes
0answers
374 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 ...
-1
votes
1answer
191 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 ...
4
votes
1answer
122 views

Simpler derivation of $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$ [duplicate]

I know that the equality $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$ can be proved in numerous ways by using the Fourier series. However, is there a way to derive it using more fundamental ...
0
votes
1answer
213 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 ?
-1
votes
1answer
137 views

Proof of $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$ [duplicate]

Prove that $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$ I have actually got the HINT for this Proof from a very nice book: The HINT i started with is: $$\frac{1}{\sin^2 x}=\frac{1}{4\sin^2\...
1
vote
0answers
159 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 ...
-3
votes
3answers
121 views

Sum of $ 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+…$ [duplicate]

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....

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