For consider the two series A(p)= _ n=1 ^ 1 n^ p , B(p)= _ n=1 ^ (-1)^ n+1 1 n^ p

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

Problem

For

p>0

consider the two series

A(p)=\sum_{n=1}^{\infty} \frac1{n^{p}}, \qquad B(p)=\sum_{n=1}^{\infty} (-1)^{n+1}\frac1{n^{p}}.

Take

p=0.99

in what follows. (a) Use a suitable comparison to determine whether

A(0.99)

converges or diverges. Show all details. (b) Determine whether

B(0.99)

converges or diverges. Present a logically ordered argument based on properties of its terms. (c) Classify the convergence of

B(0.99)

as absolute or conditional, and justify your classification with explicit reference to your conclusions in parts (a) and (b). (d) Suppose

S_N

is the

N

th partial sum of

B(0.99)

and

R_N=S-S_N

is the remainder. Provide a general inequality that gives an explicit numerical bound for

|R_N|

in terms of

N

, and explain why your bound is valid.