Let T(a,b)= _ n=1 ^ (-1)^ n , 1 n^ a +n^ b

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

Problem

Let

T(a,b)=\sum_{n=1}^{\infty}(-1)^{n}\,\frac{1}{n^{a}+n^{b}},\qquad a,b\in\mathbb R.

The convergence of the series depends on the ordered pair

(a,b)

. (a) Evaluate

\displaystyle\lim_{n\to\infty}\dfrac{1}{n^{a}+n^{b}}

in terms of

a

and

b

. Express the limit using the dominant power of

n

. (b) Determine precisely the set of all

(a,b)

for which the limit in part (a) equals

0

. Present your answer as a region in the

ab

–plane and justify how you obtained it. (c) For ordered pairs

(a,b)

in the region found in part (b), prove that the sequence

c_n(a,b)=\dfrac{1}{n^{a}+n^{b}}

is eventually decreasing. Your argument should treat all possible orderings of

a

and

b

systematically. (d) Combine your work from parts (b) and (c) to state and justify the full set of

(a,b)

for which the series

T(a,b)

converges.