Mathematics

Problem

The series

\textstyle S=\sum_{n=1}^{\infty} \frac{1}{n\sqrt{n+2}}

is to be analyzed in comparison with the familiar

p

-series

P=\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}.

Let

a_n=\dfrac{1}{n\sqrt{n+2}}

and

b_n=\dfrac{1}{n^{3/2}}

for

n\ge 1

. a. Part 1: Verify that

a_n>0

and

b_n>0

for all

n\ge 1

. Briefly explain why this fact is essential for the argument that follows. b. Part 2: Compute the limit

L=\lim_{n\to\infty}\frac{a_n}{b_n}.

Show every algebraic step leading to the numerical value of

L

. Hint: rationalize or rewrite radicals as needed to evaluate the limit without invoking series tests by name. c. Part 3: Using the value of

L

obtained in part (b) together with the known behavior of

P

, determine whether

S

converges or diverges. Provide a clear justification that explicitly connects the limit from part (b) to your conclusion.