For consider the family of series S_p= _ n=1 ^ 5 n^ p +n

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

Problem

For

p>0

consider the family of series

S_p=\sum_{n=1}^{\infty} \frac{5}{n^{p}+n}.

Let

a_n^{(p)}=\dfrac{5}{n^{p}+n}

denote the

n

-th term of

S_p

. (1) For

p>0

evaluate the limit

L(p)=\lim_{n\to\infty}\frac{a_n^{(p)}}{\dfrac{1}{n^{p}}}

and simplify the result so that it is expressed solely in terms of

p

. Show each algebraic transformation, including the treatment of different possible values of

p

. (2) Using the limit obtained in part (a) and the known behavior of

\sum \dfrac{1}{n^{p}}

, determine precisely for which values of

p

the series

S_p

converges and for which values it diverges. Justify every case in your answer.