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}

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
asked 2021-02-02
Which series converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
\(\sum_{n=1}^\infty\frac{2}{10^n}\)

Answers (1)

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: \(\frac{2}{9}\)
0

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