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

The sum from n=0 to infinity of 2(2/3)^n is 6

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

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

6

Solution steps

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

Apply the constant multiplication rule:

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

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

Apply Series Geometric Test:

3

= 2 \cdot 3

Simplify

= 6

n = 1 \sum \infty \frac{n}{2}

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What is the sum from n=0 to infinity of 2(2/3)^n ?
The sum from n=0 to infinity of 2(2/3)^n is 6