- $\displaystyle \displaystyle \sum _{n=1}^{\infty }\left[\frac{1}{n^a}-\frac{1}{(n+1)^a}\right],\ a\in \mathbb{R}$
- $\displaystyle \displaystyle \sum _{n=1}^{\infty }\ln \left(1+\frac{1}{n}\right).$

How would I go about finding whether these series are convergent or divergent using the definition (ie. the series i=1 to infinity of ai is convergent if the limit as n approaches infinity of the series i=1 to n of ai exists, otherwise it is divergent)

any help would be appreciated thanks