Mathematics

Problem

Define the series

T=\sum_{n=5}^{\infty}\frac{n+2}{n^{3/2}}.

Each term is positive, and the index of summation begins at

n=5

. Complete the tasks below. (a) Express the general term as a single power of

n

times a factor that remains bounded as

n\to\infty

. (b) Establish a numerical inequality that compares the general term of

T

with the corresponding term of a well-known divergent series for all

n\ge N

(specify

N

). (c) Using part (b), decide whether

T

converges or diverges. Justify your conclusion with clear reasoning.