Determine the minimal number of terms required to approximate the sum of the series with an error of less than 0.004. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ! }$
A) 10
B) 5
C) 8
D) 6
E) 4

Question

Quizplus · Accepted Answer

The series is the Taylor series expansion of $e^{-1}$ around 0. To find the minimal number of terms needed for an approximation with an error less than 0.004, we can use the remainder term of the Taylor series, which is given by the next term in the series. For the series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ! }$, the (n+1)th term is $\frac{1}{(n+1)!}$. We need to find the smallest n such that $\frac{1}{(n+1)!} < 0.004$. Starting from n=0, we find that for n=5, $\frac{1}{6!} = \frac{1}{720} < 0.004$, so 6 terms are required to ensure the error is less than 0.004.

Determine the Minimal Number of Terms Required to Approximate the Sum