Find the infinite sum accurate to three decimal places.
-$$\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 1 } { n ^ { 5 } }$$
A) -0.972
B) 0.972
C) 1.036
D) -1.036

Question

Quizplus · Accepted Answer

This series is a convergent alternating series, specifically a form of the Dirichlet eta function, which for $s=5$ converges to a positive value. The alternating signs $(-1)^{n+1}$ ensure the series alternates between positive and negative terms, but since the series starts with $n=1$, the first term is positive. The convergence to a specific value accurate to three decimal places requires calculation or lookup, but the key is recognizing the series converges to a positive value, hence option B (0.972) is correct.

Find the Infinite Sum Accurate to Three Decimal Places ∑n=1∞(−1)n+11n5\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 1 } { n ^ { 5 } }n=1∑∞​(−1)n+1n51​

Find the Infinite Sum Accurate to Three Decimal Places $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 1 } { n ^ { 5 } }$