`f(x) = (4x)/(x^2+2x-3) ,c=0` Find a power series for the function, centered at c and determine the interval of convergence.

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` f(x)=(4x)/(x^2+2x-3), c=0`

Let's first factorize the denominator of the function,




Now let, `(4x)/(x^2+2x-3)=A/(x+3)+B/(x-1)`




equating the coefficients of the like terms,

`A+B=4`    ----------------(1)

`-A+3B=0`   ------------(2)

From equation 2,


Substitute A in equation 1,




Plug in the value of B in equation 2,



The partial fraction decomposition is thus,




Since both fractions are in the form of `a/(1-r)`

Power series is a geometric series,




Interval of convergence `|-x/3|<1,|x|<1`

`|x/3|<1`  and `|x|<1`

`-3<x<3` and `-1<x<1`

Interval of convergence is the smaller of the intervals of convergence of the two individual fractions,

So, Interval of convergence is (-1,1)


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