What is the sum from n=1 to infinity of (ne^n)/(n!) ?

The sum from n=1 to infinity of (ne^n)/(n!) is converges

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sum from n=1 to infinity of (ne^n)/(n!)

n = 1 \sum \infty \frac{n e ^{n}}{n !}

converges

Solution steps

n = 1 \sum \infty \frac{n e ^{n}}{n !}

Apply Series Ratio Test:

converges

= converges

n = 1 \sum \infty \frac{n}{3 n}

n = 0 \sum \infty \frac{n}{2 n ^{3} + 1}

n = 0 \sum \infty n

n = 1 \sum \infty (\frac{5 π}{6})^{n}

k = 1 \sum \infty \frac{5}{k}

What is the sum from n=1 to infinity of (ne^n)/(n!) ?
The sum from n=1 to infinity of (ne^n)/(n!) is converges