Question

# Find the sim of each of the following series.1) \sum_{n=1}^\infty nx^n,\ |x|<12) \sum_{n=1}^\infty \frac{n}{8^n}

Series

Find the sim of each of the following series.
1) $$\sum_{n=1}^\infty nx^n,\ |x|<1$$
2) $$\sum_{n=1}^\infty \frac{n}{8^n}$$

2021-05-24

1) Find the sum of the series $$\sum_{n=1}^\infty nx^n,\ |x|<1$$
$$\sum_{n=1}^\infty nx^n=x+2x^2+3x^3+4x^4+...$$
$$=x(1+2x+3x^2+4x^3+...)$$
$$=x\sum_{n=1}^\infty nx^{n-1}$$
$$=x\frac{1}{(1-x)^2}$$
$$=\frac{x}{(1-x)^2}$$
2) Find the sum of the series $$\sum_{n=1}^\infty\frac{n}{8^n}$$
$$\sum_{n=1}^\infty\frac{n}{8^n}=\sum_{n=1}^\infty n(\frac{1}{8})^n$$
Compare with $$\sum_{n=1}^\infty nx^n$$, then $$x=\frac{1}{8}$$
$$\sum_{n=1}^\infty nx^{n-1}=\frac{x}{(1-x)^2}$$
$$=\frac{1/8}{(1-\frac{1}{8})^2}$$
$$=\frac{1/8}{(7/8)^2}$$
$$=\frac{1/8}{49/64}$$
$$=\frac{8}{49}$$