Define a family of -series variants by V(t)= _ n=2 ^ 1 n ,( n)^ t

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

Problem

Define a family of

p

-series variants by

V(t)=\sum_{n=2}^{\infty}\frac{1}{n\,(\ln n)^{t}}, \qquad t\in\mathbb R.

Answer the following. (a) Show that the sequence

v_n=\dfrac{1}{n(\ln n)^{t}}

is positive and decreasing for all

n\ge 3

when

t>0

. (b) Determine all real values of

t

for which the series

V(t)

converges. (c) For the boundary value(s) of

t

identified in part (b), analyze the behavior of

V(t)

and state whether it converges or diverges, giving complete justification.