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Question 8.3.3: Estimating the Error in a Partial Sum Estimate the error in ...

Estimating the Error in a Partial Sum

Estimate the error in using the partial sum S_{100} to approximate the sum of the series \sum_{k=1}^{\infty} \frac{1}{k^3} .

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First, recall that in example 3.2, we used the Integral Test to show that this
series (a p-series, with p = 3) is convergent. From Theorem 3.2, the remainder satisfies

0 \leq R_{100} \leq \int_{100}^{\infty} \frac{1}{x^3} d x=\lim _{R \rightarrow \infty} \int_{100}^R \frac{1}{x^3} d x=\lim _{R \rightarrow \infty}\left(-\frac{1}{2 x^2}\right)_{100}^R

=\lim _{R \rightarrow \infty}\left(\frac{-1}{2 R^2}+\frac{1}{2(100)^2}\right)=5 \times 10^{-5} \text {. }

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