# Determining if a series is absolutely convergent or conditionally convergent without the usage of the limit comparison test

Would the following series be conditionally convergent, absolutely convergent or divergent?

$$\sum^\infty_{k=1}\frac{k\sin{(1+k^3)}}{k+\ln{k}}$$

Whereas for sine functions in series like this, you can usually say that it just alternates between 1 and -1 but would this one do the same for $k\geq1$? I don't think it would so that would mean I can't use the alternating series test. Perhaps, then, the ratio test or some comparison test?

• You know this question wasn't solved though. – AustereTiger Mar 19 '18 at 16:48
• @AusterTiger Did you verify with your instructor? – gimusi Mar 19 '18 at 16:57
• I can't. This question is in a test. – AustereTiger Mar 19 '18 at 18:04

## 1 Answer

HINT

The strategy is to show that the necessary condition doesn't hold since $a_k \not \to 0$ and thus that $\sum a_k$ diverges.

• That just shows it is not absolutely convergent. It still could be conditionally convergent. – Mark Fischler Mar 15 '18 at 20:08
• @MarkFischler: What? $\sum a_n$ convergent implies that $a_n \to 0$ – Martin R Mar 15 '18 at 20:09
• @Martin R Consider the series $\sum (-1^k) (1-\frac1k)$, which is conditinally convergent. – Mark Fischler Mar 15 '18 at 20:11
• @MarkFischler It would be interesting to see a full answer from you. – gimusi Mar 15 '18 at 20:14
• @MarkFischler: Is it? A series is convergent if the sequence of partial sums $s_n = \sum_{k=1}^n a_k$ has a limit, and that implies that $a_n = s_n - s_{n-1} \to 0$. Compare en.wikipedia.org/wiki/Term_test. – Martin R Mar 15 '18 at 20:15