For lambda=3, the sum is

where all a

_{i}<1. With k -> inf, this sum vanishes, but the first terms and their sum diverge.

In the limit k->inf:

With lambda=-2, it diverges (oscillates between 0 for odd k and 2 for even k)

With -2<lambda<2, it converges to 1.

With lambda=2, it converges to 2.

With lambda>2, it diverges

With lambda< -2, I would expect divergence, too.

The series itself converges for every k>1, I agree. For every lambda, only a finite number of terms is larger than 1 and with k=1 it converges, therefore taking the k'th power does not change convergence. However, the limit k->inf depends on lambda.

No useful idea, sorry.