A power series is a series in the form

\sum_{n=0}^\infty c_n(x-a)^n

Power series can be used to represent functions in the form


since this is the formula for an infinite geometric series. For example,


over the interval of convergence, which is |x|<\frac{1}{5}

Functions can sometimes be differentiated or integrated to put them in a form where they can represented by a power series. For example,

\frac{d}{dx}\arctan(x)=\frac{1}{1+x^2}=\sum_{n=0}^\infty(-1)^n x^2n



An important application of power series is in Taylor and Maclaurin series.

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