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


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