Sum((19/20)^n, (n, 0, oo))

The calculator will try to find the sum $$$\sum_{n=0}^{\infty} \left(\frac{19}{20}\right)^{n}$$$ or tell whether it is convergent, with steps shown.

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Your Input

Find $$$\sum_{n=0}^{\infty} \left(\frac{19}{20}\right)^{n}$$$.

Solution

$$$\sum_{n=0}^{\infty} \left(\frac{19}{20}\right)^{n}$$$ is an infinite geometric series with the first term $$$b=1$$$ and the common ratio $$$q=\frac{19}{20}$$$.

By the ratio test, it is convergent.

Its sum is $$$S=\frac{b}{1-q}=20$$$.

Therefore,

$${\color{red}{\left(\sum_{n=0}^{\infty} \left(\frac{19}{20}\right)^{n}\right)}}={\color{red}{\left(20\right)}}$$

Hence,

$$\sum_{n=0}^{\infty} \left(\frac{19}{20}\right)^{n}=20$$

Answer

$$$\sum_{n=0}^{\infty} \left(\frac{19}{20}\right)^{n} = 20$$$A

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