I want to prove that the series $$ \sum_{n=1}^{\infty}(-1)^n\left(2^{1/n}-1\right)$$ converges.
I am fairly certain that this converges. Using the ratio or root test does not seem to work. Therefore the other tests I am left with are alternating series test, intergral test, comparison test and limit comparison test (that we know about).
Seems like it would be a messy integral and that would not work or I would not be able to solve it.
I think I could show this using the alternating series test IF the limit of the series is 0. My problem is formally showing that the limit of this series is 0. I can't quite get past that.
My only option left if the alternating series test does not work is the limit comparison test, since the series is not > 0 for large n, and therefore we cannot use the comparison test. If this is the case I honestly don't have a clue what other series I should compare this with.
Thank you for reading!