Power & Taylor series

A power series is a sum of terms $c_n(x-a{)}^n$. You can differentiate or integrate it term-by-term inside its interval of convergence. If $f(x)=\sum c_n(x-a{)}^n$, then $f^{\prime }(x)=\sum n{c}_n(x-a{)}^{n-1}$ and $\int f(x)dx=C+\sum \frac{c_n}{n+1}(x-a{)}^{n+1}$. This lets you build new series from known ones, such as $e^x=\sum \frac{x^n}{n!}$, $\sin x=\sum (-1{)}^k\frac{x^{2k+1}}{(2k+1)!}$, and $\cos x=\sum (-1{)}^k\frac{x^{2k}}{(2k)!}$. For an alternating series whose terms decrease in magnitude to zero, the remainder after stopping at a term is bounded by the absolute value of the next term.