For lambda=3, the sum is where all ai<1. With k -> inf, this sum vanishes, but the first terms and their sum diverge.For k->+Inf the series converges to 2.
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.The problem is to determine analytically this value.