I have the following question:
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
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?
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.