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

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

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

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

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