What is the sum from k=0 to infinity of (0.1)^k ?

The sum from k=0 to infinity of (0.1)^k is 1.11111…

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sum from k=0 to infinity of (0.1)^k

k = 0 \sum \infty (0.1)^{k}

1.11111 \dots

Solution steps

k = 0 \sum \infty 0. 1^{k}

Apply Series Geometric Test:

1.11111 \dots

= 1.11111 \dots

n = 0 \sum \infty \frac{2}{7 - 3 n}

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

n = 0 \sum \infty (1 - i)^{n}

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

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

What is the sum from k=0 to infinity of (0.1)^k ?
The sum from k=0 to infinity of (0.1)^k is 1.11111…