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

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

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

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

converges

Solution steps

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

Apply Series Ratio Test:

converges

= converges

n = 1 \sum \infty \frac{5}{n !}

n = 1 \sum \infty \frac{5}{4 ^{n}}

n = 0 \sum \infty arcsin (n)

n = 1 \sum \infty (4^{\frac{1}{3}})^{n}

n = 1 \sum \infty \frac{6}{n !}

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