### Home > APCALC > Chapter 12 > Lesson 12.1.2 > Problem12-19

12-19.

Given $f ( x ) = \frac { 1 } { 1 - x }$:

1. Write a fourth-degree Taylor polynomial, $p_4(x)$, about $x = 0$ for $f(x)$ near $x = 0$.

The original function and first few derivatives are listed below. Do you see a pattern?

$f(x)=(1-x)^{-1}\text{ }f^\prime(x)=(1-x)^{-2}\text{ }f^{\prime\prime}(x)=2(1-x)^{-3}\text{ }f^{\prime\prime\prime}(x)=6(1-x)^{-4}$

Evaluate the original function and each of the derivatives at $x = 0$.

$p(x)=1+1x+\frac{2x^2}{2!} + \frac{6x^3}{3!}+...$

2. Write $p_4(x)$ using sigma notation.

Simplify the terms in the equation given in Step 2 (a). Then write an expression for the general term. Use this in your summation.