What is the sum from k=1 to infinity of k^2e^{-k} ?

The sum from k=1 to infinity of k^2e^{-k} is converges

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sum from k=1 to infinity of k^2e^{-k}

k = 1 \sum \infty k^{2} e^{- k}

converges

Solution steps

k = 1 \sum \infty k^{2} e^{- k}

Apply Series Ratio Test:

converges

= converges

n = 1 \sum \infty n! (- e)^{- 4 n}

n = 1 \sum \infty \frac{6}{n ^{2} + 1}

n = 1 \sum \infty \frac{( - 1 + 2 ) ^{n}}{n}

n = 1 \sum \infty \frac{19}{1 + 3 ^{n}}

n = 1 \sum \infty \frac{ln ( n )}{7 n}

What is the sum from k=1 to infinity of k^2e^{-k} ?
The sum from k=1 to infinity of k^2e^{-k} is converges