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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1$\begingroup$ Writen down the two Taylor expansions, subtract. Use the expression you get to show that the difference (and therefore its square root) is "small." $\endgroup$– André NicolasMar 21, 2013 at 21:01
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1 Answer
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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.