Mathematics

Problem

Let

k

be a real parameter and consider the series

U_k=\sum_{n=1}^{\infty}\frac{n^{k}}{\sqrt{n^{k+2}+7}}.

For

n\ge 1

set

u_n^{(k)}=\dfrac{n^{k}}{\sqrt{n^{k+2}+7}}

. a. Show that, for large

n

, the term

u_n^{(k)}

can be written in the form

u_n^{(k)}\sim\frac{1}{n^{q(k)}},

where

q(k)

is an explicit function of

k

. Determine

q(k)

and justify your result through algebraic manipulation of exponents. b. Evaluate

L(k)=\lim_{n\to\infty}\frac{u_n^{(k)}}{\dfrac{1}{n^{q(k)}}}

and show that the limit is finite and non-zero for all real

k

. c. Using parts (a) and (b) together with the known behavior of

\sum \dfrac{1}{n^{q(k)}}

, determine all real values of

k

for which

U_k

converges and all values for which it diverges. Supply a thorough justification that considers every relevant interval of

k