Question 8.5.5: Using the Ratio Test ∑^∞k=0 (−1)^k k!/e^k for convergence.

The graph of the first 20 partial sums of the series seen in Figure 8.37 suggests that the series diverges. We can confirm this suspicion with the Ratio Test. We have

=\lim _{k \rightarrow \infty} \frac{(k+1) k !}{e k !}=\frac{1}{e}\lim _{k \rightarrow \infty} \frac{k+1}{1}=\infty . \quad \begin{array}{l}\text { Since }(k+1) !=(k+1) \cdot k ! \\\text { and } e^{k+1}=e^k \cdot e^1\end{array}.

By the Ratio Test, the series diverges, as we suspected.