What is the sum from n=0 to infinity of 1/3*(1/3)^n ?

The sum from n=0 to infinity of 1/3*(1/3)^n is 1/2

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sum from n=0 to infinity of 1/3*(1/3)^n

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

\frac{1}{2}

Decimal

0.5

Solution steps

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

Apply Series Geometric Test:

\frac{3}{2}

= \frac{1}{3} \cdot \frac{3}{2}

Cross-cancel:

3

= \frac{1}{2}

What is the sum from n=0 to infinity of 1/3*(1/3)^n ?
The sum from n=0 to infinity of 1/3*(1/3)^n is 1/2