What is the sum from n=0 to infinity of n^3e^{-n^2} ?

The sum from n=0 to infinity of n^3e^{-n^2} is converges

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sum from n=0 to infinity of n^3e^{-n^2}

n = 0 \sum \infty n^{3} e^{- n^{2}}

converges

Solution steps

n = 0 \sum \infty n^{3} e^{- n^{2}}

Apply Series Ratio Test:

converges

= converges

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

n = 0 \sum \infty \frac{4}{5 ^{\frac{n}{3}}}

n = 0 \sum \infty \frac{1}{e}

n = 2 \sum \infty 3^{- n}

n = 1 \sum \infty (- \frac{3}{5})^{n - 1}

What is the sum from n=0 to infinity of n^3e^{-n^2} ?
The sum from n=0 to infinity of n^3e^{-n^2} is converges