Taylor expansion, series convergence

Kinda stuck in here too, would like to get some help!

Using Taylor expansion check if series :

$$\sum_{n=1}^{\infty}\frac{1}{n}\sqrt{e^\frac{1}{n} - e^\frac{1}{n+1}}$$

converges.

How do I approach this kind of exercise? Thanks in advance!

• Writen down the two Taylor expansions, subtract. Use the expression you get to show that the difference (and therefore its square root) is "small." – André Nicolas Mar 21 '13 at 21:01

Hint with $e^x\sim_0 1+x+\frac{x^2}{2}$ we have $$e^\frac{1}{n} - e^\frac{1}{n+1}\sim\frac{1}{n^2}$$ so $$\frac{1}{n}\sqrt{e^\frac{1}{n} - e^\frac{1}{n+1}}\sim\frac{1}{n^2}$$ hence we have the convergence of the series by comparaison with Riemann series.