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!
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Sign up to join this communityKinda 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!
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.