# Which series converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. sum_{n=1}^inftyfrac{2}{10^n}

Question
Series
Which series converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
$$\sum_{n=1}^\infty\frac{2}{10^n}$$

2021-02-03
$$\sum_{n=1}^\infty\frac{2}{10^n}$$
$$=2\sum_{n=1}^\infty\frac{1}{10^n}$$
$$=2\sum_{n=1}^\infty(\frac{1}{10})^n$$
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: $$\frac{2}{9}$$

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