# How to find the nature of this series?

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?

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.
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.