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

The sum from k=0 to infinity of (2/3)^k is 3

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

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

3

Solution steps

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

Apply Series Geometric Test:

3

= 3

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

k = 1 \sum \infty \frac{1}{k ^{2} + 5 k + 6}

n = 1 \sum \infty 3 (0.1)^{n - 1}

n = 1 \sum \infty 6 (0.4)^{n - 1}

n = 2 \sum \infty \frac{9}{n ln ( n )}

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