Determine how many terms are needed to estimate to within 0.0001.
A) k = 30
B) k = 31
C) k = 9
D) k = 0.9999

Question

Determine how many terms are needed to estimate  to within 0.0001.
A) k = 30 
B) k = 31 
C) k = 9 
D) k = 0.9999

Quizplus · Accepted Answer

The formula for estimating e is given by the following series:

e = 1 + 1/1! + 1/2! + 1/3! + ... + 1/n!

Using the formula for the remainder of a convergent alternating series, we know that the error in approximating e by the first n terms of the series is bounded by the absolute value of the next term, namely 1/(n+1)!, divided by the sum of the terms up to that point.

So we can set up the following inequality to find the smallest value of n that makes the error less than 0.0001:

1/(n+1)! < 0.0001/(1 + 1/1! + 1/2! + ... + 1/n!)

Solving for n, we find that n = 31 is the smallest integer that satisfies this inequality. Therefore, we need 31 terms to estimate e to within 0.0001. Choice B is the correct answer.

Determine How Many Terms Are Needed to Estimate to Within