What is the sum from n=0 to infinity of 2(1/4)^n ?

The sum from n=0 to infinity of 2(1/4)^n is 8/3

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sum from n=0 to infinity of 2(1/4)^n

n = 0 \sum \infty 2 (\frac{1}{4})^{n}

\frac{8}{3}

Solution steps

n = 0 \sum \infty 2 (\frac{1}{4})^{n}

Apply the constant multiplication rule:

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

= 2 \cdot n = 0 \sum \infty (\frac{1}{4})^{n}

Apply Series Geometric Test:

\frac{4}{3}

= 2 \cdot \frac{4}{3}

Simplify

2 \cdot \frac{4}{3} : \frac{8}{3}

= \frac{8}{3}

n = 4 \sum \infty \frac{1}{n} - \frac{1}{n + 2}

n = 0 \sum \infty \frac{n}{11 n + 1}

m = 2 \sum \infty \frac{3}{4 ^{m}}

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

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

What is the sum from n=0 to infinity of 2(1/4)^n ?
The sum from n=0 to infinity of 2(1/4)^n is 8/3