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 } t ^ { n } } { n } , - 1

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.
-$$\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } t ^ { n } } { n } , - 1 < t \leq 1$$
A) $\mid$ Error $| < | \frac { t ^ { 5 } } { 5 } \mid$
B) $\mid$ Error $| < | t ^ { 5 } \mid$
C) $\mid$ Error $\mid < 0.20$
D) $\mid$ Error $| < | \frac { t ^ { 4 } } { 4 } \mid$

Quizplus · Accepted Answer

The given series is an alternating series with decreasing terms. Therefore, using the alternating series estimation theorem, the error in approximating the sum of the series by the sum of the first four terms is less than or equal to the absolute value of the next term in the series, which is $\frac{|t|^5}{5}$. Therefore, the correct choice is A.

Estimate the Magnitude of the Error Involved in Using the Sum