What is the sum from n=1 to infinity of 5n^2e^{-n^3} ?

The sum from n=1 to infinity of 5n^2e^{-n^3} is converges

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sum from n=1 to infinity of 5n^2e^{-n^3}

n = 1 \sum \infty 5 n^{2} e^{- n^{3}}

converges

Solution steps

n = 1 \sum \infty 5 n^{2} e^{- n^{3}}

Apply the constant multiplication rule:

\sum c \cdot a_{n} = c \cdot \sum a_{n}

= 5 \cdot n = 1 \sum \infty n^{2} e^{- n^{3}}

Apply Series Ratio Test:

converges

= 5 converges

= converges

What is the sum from n=1 to infinity of 5n^2e^{-n^3} ?
The sum from n=1 to infinity of 5n^2e^{-n^3} is converges