Can someone find why the following series converges?
[math]\sum_{k=1} ^{\infty} \left( e^{\frac{1}{k} } -1 - \frac{1}{k} \right) [/math]
>>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.
>>9167986
I love you anon
>>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.