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

The sum from n=0 to infinity of 4(2/3)^n is 12

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sum from n=0 to infinity of 4(2/3)^n

n = 0 \sum \infty 4 (\frac{2}{3})^{n}

12

Solution steps

n = 0 \sum \infty 4 (\frac{2}{3})^{n}

Apply the constant multiplication rule:

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

= 4 \cdot n = 0 \sum \infty (\frac{2}{3})^{n}

Apply Series Geometric Test:

3

= 4 \cdot 3

Simplify

= 12

n = 1 \sum \infty \frac{n ^{4}}{3 n !}

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

n = 0 \sum \infty 0^{n}

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

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

What is the sum from n=0 to infinity of 4(2/3)^n ?
The sum from n=0 to infinity of 4(2/3)^n is 12