Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. f(x)=frac{3}{3+x}

Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. f(x)=frac{3}{3+x}

Question
Series
asked 2021-02-03
Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.
\(f(x)=\frac{3}{3+x}\)

Answers (1)

2021-02-04
Consider the function \(f(x)=\frac{3}{3+x}\)
\(f(x)=\frac{3}{3+x}\)
\(=\frac{3}{3(1+\frac{x}{3})}\)
\(=\frac{1}{(1+\frac{x}{3})}\)
\(=(1+\frac{x}{3})^{-1}\)
Since \((1+x)^{-1}=1-x+x^2-x^3+...+(-1)^nx^n+...\) valid when |x|
\(f(x)=(1+\frac{x}{3})^{-1}\)
\(=1-\frac{x}{3}+(\frac{x}{3})^2-(\frac{x}{3})^3+...+(-1)^n(\frac{x}{3})^n+...\) valid for \(|\frac{x}{3}|<1\)</span>
\(=1-\frac{x}{3}+\frac{x^2}{9}-\frac{x^3}{27}+...+(-1)^n(\frac{x}{3})^n+...\) the expansion valid for |x|
Therefore, the power series of expansion is
\(f(x)=1-\frac{x}{3}+\frac{x^2}{9}-\frac{x^3}{27}+...+(-1)^n(\frac{x}{3})^n+...\) or \(f(x)=\sum_{n=0}^\infty(-1)^n(\frac{x}{3})^n\)
for |x|
The expansion valid for |x|
Thus, the radius of convergence is R=3.
0

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