\(\sum_{n=1}^\infty\frac{2}{10^n}\)

\(=2\sum_{n=1}^\infty\frac{1}{10^n}\)

\(=2\sum_{n=1}^\infty(\frac{1}{10})^n\)

This a geometric series with common ratio \(=\frac{1}{10}<1\)</span>

So this series will converge

Then we use the infinite geometric series sum formula

\(=2\frac{(\frac{1}{10})^1}{1-\frac{1}{10}}=2\frac{(\frac{1}{10})}{\frac{9}{10}}=\frac{2}{9}\)

Answer: \(\frac{2}{9}\)

\(=2\sum_{n=1}^\infty\frac{1}{10^n}\)

\(=2\sum_{n=1}^\infty(\frac{1}{10})^n\)

This a geometric series with common ratio \(=\frac{1}{10}<1\)</span>

So this series will converge

Then we use the infinite geometric series sum formula

\(=2\frac{(\frac{1}{10})^1}{1-\frac{1}{10}}=2\frac{(\frac{1}{10})}{\frac{9}{10}}=\frac{2}{9}\)

Answer: \(\frac{2}{9}\)