find-n-1-cos-nx-n-2-

Question Number 67384 by mathmax by abdo last updated on 26/Aug/19

f i n d \sum_{n = 1}^{\infty} \frac{c o s (n x)}{n^{2}}

Commented by mathmax by abdo last updated on 27/Aug/19

w e h a v e p r o v e d t h a t \sum_{n = 1}^{\infty} \frac{s i n (n x)}{n} = \frac{π - x}{2} \Rightarrow

\sum_{n = 1}^{\infty} \frac{1}{n} \int s i n (n x) d x = \frac{1}{2} \int (π - x) d x + c \Rightarrow

- \sum_{n = 1}^{\infty} \frac{1}{n^{2}} c o s (n x) = \frac{1}{2} {π x - \frac{x^{2}}{2}} + c = \frac{π}{2} x - \frac{x^{2}}{4} + c \Rightarrow

\sum_{n = 1}^{\infty} \frac{c o s (n x)}{n^{2}} = - \frac{π}{2} x + \frac{x^{2}}{4} - c

x = 0 \Rightarrow \sum_{n = 1}^{\infty} \frac{1}{n^{2}} = - c \Rightarrow c = - \frac{π^{2}}{6} \Rightarrow \sum_{n = 1}^{\infty} \frac{c o s (n x)}{n^{2}} = \frac{x^{2}}{4} - \frac{π x}{2} + \frac{π^{2}}{6}