= Power series
{title2=$\sum_{n=0}^\infty a_n(z-c)^n$}
{wiki}
A <power series> combines a <sequence> of <coefficients> with <monomials> in the displacement from a fixed centre:
$$
\sum_{n=0}^{\infty}a_n(z-c)^n.
$$
The variable $z$, centre $c$, and <coefficients> can be <real numbers> or <complex numbers>. Its degree-$N$ <partial sum> $P_N(z)=\sum_{n=0}^{N}a_n(z-c)^n$ is a <polynomial>, so a convergent <power series> can turn a complicated <function> into increasingly accurate finite calculations. Convergence, and the error after stopping, are essential parts of that claim.
The <geometric series> supplies the simplest computed example:
$$
\frac1{1-z}=1+z+z^2+\cdots,\qquad |z|<1.
$$
At $z=1/2$, stopping after degree ten gives $P_{10}(1/2)=1.9990234375$, differing from the exact value $2$ by $1/1024=0.0009765625$. At $z=0.99$, the same degree gives only about $10.4662$ instead of $100$. The same formula has very different practical accuracy at different points; the <geometric power-series remainder> quantifies it.
The structure is useful beyond numerical evaluation. Within its <radius of convergence>, <termwise differentiation of a power series> turns a question about a <derivative> into a question about <coefficients>. A <Taylor series> chooses $a_n=f^{(n)}(c)/n!$ when it represents the <function> locally. A <formal power series> instead uses the coefficients algebraically, without asserting numerical convergence; the existing <generating function> examples use that distinction to encode counting problems.
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