Answer
(a) $\frac{\pi^{2}-6}{6}$
(b) $\frac{6\pi^{2}-49}{36}$
(c)$ \frac{\pi^{2}}{24}$
Work Step by Step
(a) $\Sigma_{n=2}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6}-\frac{1}{1^{2}}=\frac{\pi^{2}-6}{6}$
(b) $\Sigma_{n=3}^{\infty}\frac{1}{(n+1)^{2}}=\Sigma_{n=4}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6}-1-\frac{1}{4}-\frac{1}{9}=\frac{6\pi^{2}-49}{36}$
(c) $\Sigma_{n=1}^{\infty}\frac{1}{(2n)^{2}}=\Sigma_{n=1}^{\infty}\frac{1}{4n^{2}}=\frac{1}{4} \cdot \frac{\pi^{2}}{6}=\frac{\pi^{2}}{24}$
