What is the sum from n=0 to infinity of n*(2/3)^n ?

The sum from n=0 to infinity of n*(2/3)^n is converges

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sum from n=0 to infinity of n*(2/3)^n

n = 0 \sum \infty n \cdot (\frac{2}{3})^{n}

converges

Solution steps

n = 0 \sum \infty n (\frac{2}{3})^{n}

Apply Series Ratio Test:

converges

= converges

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in v erse l a pl a ce \frac{e ^{- 2 s}}{s ^{2} + 1}

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What is the sum from n=0 to infinity of n*(2/3)^n ?
The sum from n=0 to infinity of n*(2/3)^n is converges