Consider the infinite series _ k=1 ^ 1 k^ 2 +1

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

Problem

Consider the infinite series

\sum_{k=1}^{\infty} \frac{1}{k^{2}+1}.

Let

S_n

denote the

n

-th partial sum and

R_n

the remainder

\displaystyle\sum_{k=n+1}^{\infty}\!\! 1/(k^{2}+1)

. (a) Justify that the series converges. (b) Determine the least integer

n

such that

R_n<0.01

. Show all work that leads to your answer. (c) Using the value of

n

found in part (b), state an interval of length less than

0.01

that contains the exact value of the series.