Question

In Exercises 29–34 use the ratio test to analyze whether the given series converges or diverges. If the ratio test is inconclusive, use a different test to analyze the series.

$\sum _{k=0}^{\infty }\frac{5^k}{k!}$

Short answer

The series converges.

Step-by-step solution

Step 1. Given information.

The given series is$\sum _{k=0}^{\infty }\frac{5^k}{k!}$.

Step 2. Ratio Test.

According to the given series,

$$\begin{gathered} a_{k+1}=\frac{5^{k+1}}{(k+1)!} \\ \frac{a_{k+1}}{a_k}=\frac{\frac{5^{k+1}}{(k+1)!}}{\frac{5^k}{k!}} \\ =\frac{k!5^{k+1}}{5^k(k+1)!} \\ =\frac{5}{(k+1)} \end{gathered}$$

Step 3. Take limits.

On Taking limits,

$$\begin{gathered} \underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}=\underset{k\to \infty }{\lim }\frac{5}{(k+1)} \\ =5\left(\underset{k\to \infty }{\lim }\frac{1}{(k+1)}\right) \\ =5\left(0\right)=0 \\ \text{Since,}L<1\text{,} \end{gathered}$$

Therefore, the series converges.