The sequence is defined by

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

Problem

The sequence

\bigl\{b_n\bigr\}_{n=1}^{\infty}

is defined by

b_n = \dfrac{\sqrt{n}}{n^{2}+1}

. Answer the following questions. Show all analytical steps and clearly state any theorems you invoke. (a) Compute the improper integral

\displaystyle J=\int_{1}^{\infty} \frac{\sqrt{x}}{x^{2}+1}\,dx

. (b) Determine whether the series

\displaystyle \sum_{n=1}^{\infty} b_n

converges or diverges. Support your conclusion with appropriate reasoning that references the integral calculated in part (a). (c) Let

S=\sum_{n=1}^{\infty} b_n

and

S_{10}=\sum_{n=1}^{10} b_n

. Using an integral estimate, obtain an upper bound for the remainder

|S-S_{10}|

and verify whether this bound is less than

0.01