How many terms of the alternating series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } n ^ { - 2 }$ must we add in order to be sure that the partial sum $S _ { n }$ is within 0.0001 of the sum $S$ ?
A)10
B)300
C)30,000
D)30
E)3
F)10,000
G)100
H)1000

Question

Quizplus · Accepted Answer

The series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } n ^ { - 2 }$ is an alternating series where the terms decrease in absolute value to 0. The error in approximating the sum $S$ by $S_n$ is less than the first omitted term. To ensure the error is within 0.0001, we find $n$ such that $\frac{1}{(n+1)^2} < 0.0001$. Solving $\frac{1}{(n+1)^2} < 0.0001$ gives $n+1 > 100$, so $n > 99$. Thus, at least 100 terms are needed.

How Many Terms of the Alternating Series ∑n=1∞(−1)n+1n−2\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } n ^ { - 2 }∑n=1∞​(−1)n+1n−2

How Many Terms of the Alternating Series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } n ^ { - 2 }$