Question

Find a power series for the function, centered at c. g(x) = 4x x2 2x − 3 , c = 0

Answer 1

The power series for given function $g(x)=\frac{4x}{(x-1)(x+3)}$ is $g(x)=\sum{_{n=0}^\infty}~x^n(-1+(-\frac{x}{3} )^n)$

For given question,

We have been given a function g(x) = 4x / (x² + 2x - 3)

We need to find a power series for the function, centered at c, for c = 0.

First we factorize the denominator of function g(x), we have:

$\Rightarrow g(x)=\frac{4x}{(x-1)(x+3)}$

We can write g(x) as,

$\Rightarrow g(x)=\frac{1}{x-1}+\frac{3}{x+3}\\\\\Rightarrow g(x)=\frac{-1}{1-x}+\frac{1}{1+\frac{x}{3} }\\\\\Rightarrow g(x)=\frac{-1}{1-x}+\frac{1}{1-(-\frac{x}{3} )}$

We know that, $\frac{1}{1-x}=\sum{_{n=0}^\infty}~{x^n}$ if |x| < 1

and $\frac{1}{1-(-\frac{x}{3} )}=\sum{_{n=0}^\infty}~x^n(-\frac{x}{3} )^n$  if $|\frac{x}{6}| < 1$

$\Rightarrow g(x)=-\sum{_{n=0}^\infty}~x^n+\sum{_{n=0}^\infty}~x^n(-\frac{x}{3} )^n$     if |x| < 1 and  if $|\frac{x}{6}| < 1$

$\Rightarrow g(x)=\sum{_{n=0}^\infty}~x^n(-1+(-\frac{x}{3} )^n)$ if |x| < 1

Therefore, the power series for given function $g(x)=\frac{4x}{(x-1)(x+3)}$ is $g(x)=\sum{_{n=0}^\infty}~x^n(-1+(-\frac{x}{3} )^n)$

Learn more about the power series here:

brainly.com/question/11606956

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