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

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

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

k = 1 \sum \infty \frac{k !}{k ^{k}}

converges

Solution steps

k = 1 \sum \infty \frac{k !}{k ^{k}}

Apply Series Ratio Test:

converges

= converges

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

n = 2 \sum \infty (\frac{3}{8})^{2 n}

n = 1 \sum \infty \frac{7}{π ^{n}}

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

n = 1 \sum \infty \frac{1}{n + 1} - \frac{1}{n}

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