How to Use Maclaurin Expansion to Find e Correct to Four Decimal Places?

[pat666]

Homework Statement

Use the Maclaurin expansion of e^x to find the value of e correct to four decimal places. (This is not the same as simply using the first four terms of the expansion.)

I did the question but i had to look up how many terms to use to be accurate to four decimal places (8) so I am wondering if i should have done it a different way?

Homework Equations

The Attempt at a Solution

see the attachment please.

 

[Gib Z]

To make sure you are accurate to 4 decimal places, you have to be sure all the terms you leave out are less than 0.00005, so that leaving it out doesn't change the 4th decimal place even with rounding.

To do that, you need to estimate the size of the terms you leave you, and this is usually done with a remainder term:

e= 1+ 1+ \frac{1}{2!} ... + \frac{1}{k!} + \frac{c}{(k+1)!}

where c is some number between 0 and 1.
So the size of the last term is certainly less than if we take c=1, ie less than 1/(k+1)!. Now you just need to find the value of k so that the last term is less than 0.00005.