A power series $$\sum_{n=0}^\infty a_n x^n$$ can be thought of as a function of $x$ that is defined inside the interval of convergence. Not all functions can be expressed as power series, but most common and useful functions can.
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Example: Since
$$1+r+r^2+r^3+ \ldots = \frac{1}{1-r},$$
for $\lvert r\rvert<1$, replacing $r$ with $x$ gives us the first example of a function expressed as power series, namely
$$
\displaystyle{f(x)=\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1+x+x^2+\ldots}$$ as long as $\lvert x\rvert<1$.
Similarly, replacing $r=-x$ gives us $$\displaystyle{\frac{1}{1+x} = \sum_{n=0}^\infty (-x)^n = 1-x+x^2+\ldots}$$ as long as $\lvert x\rvert<1$. and replacing $r=x^2$ yields $$\displaystyle{\frac{1}{1+x^2} = \sum_{n=0}^\infty (-x^2)^n = 1-x^2-x^4+x^6+\ldots}$$ as long as $|x^2|<1$. That is, as long as $\lvert x\rvert<1$. |