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

4 e^{2 x^{2} - 4}

- \frac{3}{2} x^{2} + 9 x + 9

\int (12 x^{3} - 4 x - 2) d x

\int \frac{1}{\frac{1}{v}} d v

\frac{e ^{- 2 s}}{s ^{2} + 1}

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