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Can someone find why the following series converges?

[math]\sum_{k=1} ^{\infty} \left( e^{\frac{1}{k} } -1 - \frac{1}{k} \right) [/math]

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>>9167956

[math]e^x=1+x+o(x^2)\quad (x\to 0)[/math]

[math]\implies a_k=e^{\frac{1}{k}}-1-\frac{1}{k}=o(\frac{1}{k^2})\quad (k\to \infty)[/math]

Because [math]\sum\frac{1}{k^2}[/math] converges, [math]\sum a_k[/math] converges.

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>>9167986

I love you anon

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>>9167986

Small error, it should be

[math]e^x=1+x+x^2+o(x^2)\quad (x\to 0)[/math]

[math]\implies a_k=e^{\frac{1}{k}}-1-\frac{1}{k}=\frac{1}{k^2} + o(\frac{1}{k^2})\quad (k\to \infty)[/math]

Because [math]\sum\frac{1}{k^2}[/math] converges, [math]\sum a_k[/math] converges.

Sorry.

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