Ask an Expert
Question

# Differentiate the power series for f(x)=xe^x. Use the result to find the sum of the infinite series sum_{n=0}^inftyfrac{n+1}{n!}

Series
ANSWERED
asked 2021-03-02
Differentiate the power series for $$f(x)=xe^x$$. Use the result to find the sum of the infinite series
$$\sum_{n=0}^\infty\frac{n+1}{n!}$$

## Answers (1)

2021-03-03
Consider the function $$f(x)=xe^x$$
Differentiate the above function to get,
$$f'(x)=\frac{d}{dx}(xe^x)$$
$$=xe^x+e^x$$
$$=\sum_{n=0}^\infty\frac{(n+1)x^n}{n!}$$
Substitute 1 for x in the above series,
$$1e^1+e^1=\sum_{n=0}^\infty\frac{(n+1)(1)^n}{n!}$$
$$2e=\sum_{n=0}^\infty\frac{(n+1)}{n!}$$
$$\sum_{n=0}^\infty\frac{(n+1)}{n!}\approx5.4363$$
Thus, the sum of the infinite series is approximately 5.436.

expert advice

...