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# Can someone find why the following series converges? $\sum_{k=1} This is a blue board which means that it's for everybody (Safe For Work content only). If you see any adult content, please report it. Thread replies: 4 Thread images: 1 File: IMG_2111.jpg (1MB, 3264x2448px) Image search: [iqdb] [SauceNao] [Google] 1MB, 3264x2448px Can someone find why the following series converges? [math]\sum_{k=1} ^{\infty} \left( e^{\frac{1}{k} } -1 - \frac{1}{k} \right)$
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>>9167956
$e^x=1+x+o(x^2)\quad (x\to 0)$
$\implies a_k=e^{\frac{1}{k}}-1-\frac{1}{k}=o(\frac{1}{k^2})\quad (k\to \infty)$
Because $\sum\frac{1}{k^2}$ converges, $\sum a_k$ converges.
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>>9167986
I love you anon
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>>9167986
Small error, it should be
$e^x=1+x+x^2+o(x^2)\quad (x\to 0)$
$\implies a_k=e^{\frac{1}{k}}-1-\frac{1}{k}=\frac{1}{k^2} + o(\frac{1}{k^2})\quad (k\to \infty)$
Because $\sum\frac{1}{k^2}$ converges, $\sum a_k$ converges.

Sorry.