Determine whether the infinite series is convergent or not

[Kemikeren]

I have the following infinite series:



How do I proceed to determine whether it is convergent or not?

 

[HallsofIvy]

I have the following infinite series:

View attachment 10204

How do I proceed to determine whether it is convergent or not?

I would be inclined to use the "quotient test". \(\displaystyle \sum a_n\) converges if \(\displaystyle \lim_{n\to \infty} \left|\frac{a_{n+1}}{a_n}\right|< 1\) and diverges if that limit is greater than 1.

 

[lookagain]

I already made a reply post after HallsofIvy's. Why is it (still) not here!?

This just a summary:

Again, the ratio test is inconclusive. It comes out that the limit as n approaches
infinity equals i (or some multiple of it).

Writing out several terms shows that the series is equivalent to A + Bi, where
A and B are each alternating series which are also convergent.