Question

Explain why the series $\sum _{k=0}^{\infty }\frac{1}{k!}$converges. Which convergence tests could be used to prove this?

Short answer

Hence proved.

Step-by-step solution

Step 1. Given Information.

The given series is$\sum _{k=0}^{\infty }\frac{1}{k!}$.

Step 2. Ratio Test.

Using the ratio test,

$$\begin{gathered} \frac{a_{k+1}}{a_k}=\frac{\frac{1}{(k+1)!}}{\frac{1}{k!}} \\ \frac{a_{k+1}}{a_k}=\frac{k!}{(k+1)k!} \\ =\frac{1}{k+1} \\ \underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}=\underset{k\to \infty }{\lim }\frac{1}{(k+1)} \\ =1\left(\underset{k\to \infty }{\lim }\frac{1}{(k+1)}\right) \\ =0 \\ Since,L<1, \end{gathered}$$

Therefore, the series converges.