Mathematics

Problem

Two series are defined below: Series A:

\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{3}}

Series B:

\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{3}+n}

Answer each part, providing clear reasoning and appropriate justification. (a) Evaluate each of the following improper integrals. (i)

\displaystyle \int_{1}^{\infty} \frac{1}{x^{3}}\,dx

(ii)

\displaystyle \int_{1}^{\infty} \frac{1}{x^{3}+x}\,dx

(b) For each series (A and B), state whether it converges or diverges and justify your answer using your calculations from part (a). (c) Suppose

S_A

and

S_B

denote the sums of Series A and Series B, respectively, and

S_{A,N}

and

S_{B,N}

their partial sums through index

N

. Based on your work in parts (a) and (b), indicate which series’ partial sums approach their finite limit more rapidly as

N

grows large. Provide a brief explanation referencing the sizes of the corresponding improper integrals that bound the remainders.