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}

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asked 2021-05-23

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

Answers (1)

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 \(x=\frac{1}{8}\)
\(\sum_{n=1}^\infty nx^{n-1}=\frac{x}{(1-x)^2}\)
\(=\frac{1/8}{(1-\frac{1}{8})^2}\)
\(=\frac{1/8}{(7/8)^2}\)
\(=\frac{1/8}{49/64}\)
\(=\frac{8}{49}\)

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