Science:Math Exam Resources/Courses/MATH101/April 2018/Question 09 (ii)/Statement

Harder: Consider the infinite series

$$\displaystyle \sum _{n=1}^{\infty }{\frac {\left({\sqrt {1+n}}-{\sqrt {n}}\right)}{n^{p}}}\,.$$

Find, with a detailed explanation, the range of ${\textstyle p}$ for which the infinite series converges.

Hint: One approach to this problem is to think about the numerator for large ${\textstyle n}$.