- #1

JG89

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**1. The problem \statement, all variables and given/known data**

Estimate the error involved in using the first n terms for the function [tex] F(x) = \int_0^x e^{-t^2} dt [/tex]

## Homework Equations

## The Attempt at a Solution

I am using the Lagrange form of the remainder. I need to know the n+1 derivative of e^(-t^2) but I found that the derivatives get more complicated as n increases, so I dropped that idea.

I rewrote the function like this:

[tex] \int_0^x e^{-t^2} dt = \int_0^x 1 - t^2 + \frac{t^4}{2!} - \frac{t^6}{3!} + ... + \frac{(-1)^n t^{2(n-1)}}{n!} + R_n [/tex]

I can easily integrate each term, and then the integral from 0 to x of R_n would be the remainder. Note that R_n is the remainder term for e^(-t^2), so right now I am trying what R_n is.

Rewriting e^(-t^2) as its Taylor series, I differentiated term by term for the first, second, third, etc derivative to see if I can find a general expression for the nth derivative, but I can't find one.

Any suggestions?

EDIT: I was also thinking that if there is a constant value M such that all derivatives of F are bounded above by M, and I was able to find this value M, then there would be no problem obtaining an expression to estimate the remainder.

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