1
$\begingroup$

What tests to use for this series: $$\sum_{m=1}^{\infty}(-1)^{m-1}\left(1+\frac{8}{m}\right)^m$$ I've tried alternating test and ratio test but were inconclusive. Can I apply nth term test to the absolute value of the series?

$\endgroup$
0
3
$\begingroup$

You should recognize the form $$\left(1+\frac xn\right)^n$$ as it is a widely known property (sometimes even definition) of $e$, as$$\lim_{n\to\infty} \left(1+\frac xn\right)^n=e^x$$

Thus, the terms of your summation tend to $\pm e^8$ as $m$ approaches infinity. Since the terms in your series need to converge to $0$ for the series to converge, this series is divergent.

$\endgroup$
3
$\begingroup$

It is easy to see that for all $m\ge 1$, we have

$$\left(1+\frac8m\right)^m>1$$

Since the terms of the series do not approach $0$, then the series diverges.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.