Series

Constructing a series with any assigned sum
Construct a sequence $(s_n)$ converging to the assigned sum $s$ , and consider the series

$s_0+(s_1-s_0)+(s_2-s_1)+\cdots+(s_n-s_{n-1})+\cdots$

Since its $n^{th}$ partial sum is $s_n$ , the series is convergent and has the sum $s$ .
Construct a series with a closed form, by taking any sequence converging to $0$ , and writing $s_n=s-x_n, \, n=0,1,2\dots$ .

Further reading
K. Knopp – Theory and applications of infinite series