Let [math]f[/math] be a function with derivatives of all orders throughout some interval containing [math]a[/math] as an interior point. Then the Taylor series generated by [math]f[/math] at [math]x=a[/math] is[br] [math]\sum_{k=0}^{\infty}\frac{f^{\left(k\right)}\left(a\right)}{k!}\left(x-a\right)^k=f\left(a\right)+f'\left(a\right)\left(x-a\right)+\frac{f''\left(a\right)}{2!}\left(x-a\right)^2+\cdots+\frac{f^{\left(n\right)}}{n!}\left(x-a\right)^n+\cdots[/math][br]The Taylor polynomial of order [math]n[/math] generated by [math]f[/math] at [math]x=a[/math] is[br] [math]P_n\left(x\right)=f\left(a\right)+f'\left(a\right)\left(x-a\right)+\frac{f''\left(a\right)}{2!}\left(x-a\right)^2+\cdots+\frac{f^{\left(n\right)}}{n!}\left(x-a\right)^n[/math]

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]