Hi,

I have the following question:

Part d

Using appropriate series convergence tests, prove that the series

converges if and only if the real number satisfies

So my current working is this:

Let , then

Now as tends to infinity, will tend to .

Therefore, by the ratio test:

If , the series converges

If , the series diverges

If ,

In part b of this same question, we had to prove that diverges, which I did successfully.

According to my working then, the series converges if , which only partly satisfies the requirement:

Can someone spot what I've done wrong here?

Thanks

Note: In part c of the same question, we had to state and prove the Alternating Series Test, which I did successfully, so I'm thinking that we're meant to apply it here somehow, but I'm not sure how.