Question

The Riemann Zeta Function The function $\zeta$ defined by $$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}$$ where $s$ is a complex number, is called the Riemann zeta function. Leonhard Euler was able to calculate the exact sum of the $p$ -series with $p=2$ : $$\zeta(2)=\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}$$ Use this fact to find the sum of each series. (a) $\sum_{n=2}^{\infty} \frac{1}{n^{2}}$ (b) $\sum_{n=3}^{\infty} \frac{1}{(n+1)^{2}}$ (c) $\sum_{n=1}^{\infty} \frac{1}{(2 n)^{2}}$

          The Riemann Zeta Function The function $\zeta$ defined by $$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}$$ where $s$ is a complex number, is called the Riemann zeta function.
 Leonhard Euler was able to calculate the exact sum of the $p$ -series with $p=2$ :
$$\zeta(2)=\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}$$
Use this fact to find the sum of each series.
(a) $\sum_{n=2}^{\infty} \frac{1}{n^{2}}$
(b) $\sum_{n=3}^{\infty} \frac{1}{(n+1)^{2}}$
(c) $\sum_{n=1}^{\infty} \frac{1}{(2 n)^{2}}$

Added by Marisa M.

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