A table of partial sums for the series is shown below

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

A table of partial sums for the series ∑n=2∞1n2ln⁡n\displaystyle \sum_{n=2}^{\infty}\frac{1}{n^{2}\ln n} is shown below. Values are rounded to six decimal places.
nSnS_{n}
20.240227
30.373560
40.432894
50.466829
60.488051
70.502650
80.513360
90.521558
100.528042
Answer the following questions. (a) Provide a justification that the series converges. (b) Develop an inequality that can be used to bound the remainder RN=S−SNR_{N}=S-S_{N}, where SS is the exact sum. Your inequality should be suitable for finding NN such that ∣RN∣<0.001|R_{N}|<0.001. Clearly state any comparison or estimation you employ. (c) Use your inequality from part (b) to find a specific integer NN for which ∣RN∣<0.001|R_{N}|<0.001 must hold. (d) Compute (or describe how you would compute) SNS_{N} using technology and state an approximation of the series sum that is guaranteed to be within 0.0010.001 of the exact value.

Related theory

Browse the problem library