What is the sum from n=1 to infinity of (2/5)^n ?

The sum from n=1 to infinity of (2/5)^n is 2/3

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sum from n=1 to infinity of (2/5)^n

n = 1 \sum \infty (\frac{2}{5})^{n}

\frac{2}{3}

Decimal

0.66666 \dots

Solution steps

n = 1 \sum \infty (\frac{2}{5})^{n}

n = k \sum \infty (a_{n}) = n = 0 \sum \infty (a_{n}) - a_{0} - a_{1} - \dots a_{k - 1}

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

n = 0 \sum \infty (\frac{2}{5})^{n} = \frac{5}{3}

= \frac{5}{3} - (\frac{2}{5})^{0}

Simplify

\frac{5}{3} - (\frac{2}{5})^{0} : \frac{2}{3}

= \frac{2}{3}

What is the sum from n=1 to infinity of (2/5)^n ?
The sum from n=1 to infinity of (2/5)^n is 2/3