# Does the complex series converge or diverge?

The complex series is $$\sum_{k=1}^{\infty} (i^k-\frac{1}{k^2})$$ I know the answer is that the series diverges.

Is this because $$\lim_{k\to\infty} (i^k-\frac{1}{k^2})$$ Does not exist since $$i^k$$ repeats itself infinitely? So it diverges by the kth term test for divergence?

• Yes, because $|i|=1$, so the corresponding geometric series won't converge. Or the Test for Divergence, as you stated. – Integrand Mar 26 at 2:26
• @Integrand Is it true that the sum of a divergence series, with a convergent series always diverges, and could i say, it is the sum of a divergent geometric series, with a convergent p-series? – user736276 Mar 26 at 2:28
• Yep. If your series converged, you could split it up into two convergent series. Alas, one diverges and one converges, so yours diverges. – Integrand Mar 26 at 2:30