Suppose the function f has derivatives of all orders on an interval centered at the point a. The [b]Taylor series for [i]f[/i] centered at [i]a[/i] [/b]is[br][math]f\left(a\right)+f'\left(a\right)\left(x-a\right)+\frac{f'\left(a\right)}{2!}\left(x-a\right)^2+\frac{f^{\left(3\right)}\left(a\right)}{3!}\left(x-a\right)^3+.....[/math][br][math]=\sum_{k=0}^{\infty}\frac{f^{\left(k\right)}\left(a\right)}{k!}\left(x-a\right)^k[/math][br]A Taylor series centered at 0 is called a [b]MacLaurin Series[/b]

Let [i]f[/i] have derivatives of all orders on an open interval I containing a. The Taylor series for [i]f[/i] centered at a converges to [i]f[/i], for all x in I,if and only if [math]lim_{n\longrightarrow\infty}R_n\left(x\right)=0[/math] for all x in I, where[br][math]R_n\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}\left(x-a\right)^{n++1}[/math][br]is the remainder at x (with c between x and a)