Consider the series

Use Gauss’ test (from the previous exercise, Section 10.16, Exercise #17) to prove that the series converges for and diverges for .

*Proof.* From the definition of the series we have the th and st terms,

Therefore,

Using the Taylor expansion,

So we have,

where is bounded. Letting we apply Gauss’ test to conclude that converges if and diverges if

your definition of a_n appears to be wrong, you wrote the same for a_n and a_n+1