# Using power series remainder term

morenogabr

## Homework Statement

(For power series about x=1) Using the error formula, show that $$\left|ln(1.5)-p_{3}(1.5)\right|\leq\frac{(0.5)^{4}}{4}$$

## Homework Equations

$$p_{3}(x) = x-1 - \frac{(x-1)^{2}}{2} + \frac{(x-1)^{3}}{3}$$
$$\\\epsilon_{n}(x)=\frac{f^{n+1}(\xi)}{(n+1)!}(x-x_{o})^{n+1}\\where \xi lies between x_{o} and x$$

## The Attempt at a Solution

Im sure this is an easy one, but I cant think of any useful relationship between the difference |f(x)-p_3(x)| and that piece of the remainder function... any hints?
(excuse my latex crappiness)

## Answers and Replies

CheckMate
Isn't there supposed to be a sigma notation (for sum) on the left of the first term of the second equation (power series)?

morenogabr
No, its a 3rd order polynomial that approximates the power series, and its written out term by term so sigma is uneeded. Oh, if you mean for the epsilon, its the remainder term used to measure accuracy of the approximation. Its a single term, so no sigma needed. I guess I should say something like n=3 (bc of p_3(x)) so epsi_4(x)...