Prove Convergence for series of normed space 
How would one go about proving it is convergent? I would use Dirichlet Test however, $s_n$ tends to -1.

Note that $\sum |s_n| \left\lVert v_n \right\rVert \le \sum (|s_n + 1| + 1) \left\lVert v_n \right\rVert = \sum |s_n + 1| \left\lVert v_n \right\rVert + \sum \left\lVert v_n \right\rVert$. So you can apply Dirichlet's Test and the hypothesis to the first and second sums of the last expression, respectively.
