Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.
-$$\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 0.5 ) ^ { n } } { n }$$
A) $\mid$ Error $\mid

Question

Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.
-$$\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 0.5 ) ^ { n } } { n }$$
A) $\mid$ Error $\mid < 2.60 \times 10 ^ { - 3 }$
B) $\mid$ Error $\mid < 3.13 \times 10 ^ { - 2 }$
C) $\mid$ Error $\mid < 1.56 \times 10 ^ { - 2 }$
D) $\mid$ Error $\mid < 6.25 \times 10 ^ { - 3 }$

Quizplus · Accepted Answer

The series is an alternating series, and the error in approximating the sum by the first few terms is less than the absolute value of the first omitted term. The fifth term is $\frac{(0.5)^5}{5} = 6.25 \times 10^{-3}$.

Estimate the Magnitude of the Error Involved in Using the Sum