What is the sum from n=0 to infinity of 2*(2/5)^n ?

The sum from n=0 to infinity of 2*(2/5)^n is 10/3

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sum from n=0 to infinity of 2*(2/5)^n

n = 0 \sum \infty 2 \cdot (\frac{2}{5})^{n}

\frac{10}{3}

Decimal

3.33333 \dots

Solution steps

n = 0 \sum \infty 2 (\frac{2}{5})^{n}

Apply the constant multiplication rule:

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

= 2 \cdot n = 0 \sum \infty (\frac{2}{5})^{n}

Apply Series Geometric Test:

\frac{5}{3}

= 2 \cdot \frac{5}{3}

Simplify

2 \cdot \frac{5}{3} : \frac{10}{3}

= \frac{10}{3}

What is the sum from n=0 to infinity of 2*(2/5)^n ?
The sum from n=0 to infinity of 2*(2/5)^n is 10/3