# How do you use differentiation to find a power series representation for #f(x)=1/(1+x)^2#?

Finally, differentiate this term-by-term (which is justified in the interior of the interval of convergence) to get

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To find a power series representation for ( f(x) = \frac{1}{{(1 + x)}^2} ), you can start by differentiating the geometric series representation ( \frac{1}{1 - (-x)} = \sum_{n=0}^{\infty} (-x)^n ). Then, multiply the result by ( \frac{d}{dx}(1 + x) ) and simplify. This will give you the power series representation for ( f(x) ).

The geometric series representation is ( \frac{1}{{(1 + x)}^2} = \sum_{n=0}^{\infty} (-1)^n x^n ).

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