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

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

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

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

converges

Solution steps

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

Apply the constant multiplication rule:

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

= \frac{1}{3} \cdot n = 0 \sum \infty n (\frac{2}{3})^{n}

Apply Series Ratio Test:

converges

= \frac{1}{3} converges

= converges

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