Question 8.4.5: Estimating the Sum of an Alternating Series Approximate the ...
Estimating the Sum of an Alternating Series Approximate the sum of the alternating series \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^4} by the 40th partial sum and estimate the error in this approximation.
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We leave it as an exercise to show that this series is convergent. We then
approximate the sum by
S \approx S_{40} \approx 0.9470326439.
From our error estimate (4.4), we have
\left|S-S_{40}\right| \leq a_{41}=\frac{1}{41^4} \approx 3.54 \times 10^{-7} .This says that our approximation S ≈ 0.9470326439 is off by no more than \pm 3.54 \times 10^{-7}.
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