Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.
-$$\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 0.3 ) ^ { 2 n + 1 } } { 2 n + 1 }$$
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.
-$$\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 0.3 ) ^ { 2 n + 1 } } { 2 n + 1 }$$
A) $\mid$ Error $\mid < 1.61 \times 10 ^ { - 7 }$
B) $\mid$ Error $\mid < 2.19 \times 10 ^ { - 6 }$
C) $\mid$ Error $\mid < 1.77 \times 10 ^ { - 6 }$
D) $\mid$ Error $\mid < 4.43 \times 10 ^ { - 8 }$

Quizplus · Accepted Answer

The series is an alternating series, and the error in using the sum of the first four terms to approximate the sum of the entire series is less than the magnitude of the fifth term. The fifth term is when $n = 5$, so it is $\frac{(0.3)^{2(5)+1}}{2(5)+1} = \frac{(0.3)^{11}}{11}$. Calculating this gives a value less than $1.61 \times 10^{-7}$, making option A correct.

Estimate the Magnitude of the Error Involved in Using the Sum