Chapter 11 - Infinite Series - Chapter Review Exercises - Page 591: 18

$1$

\begin{align*} \lim _{n \rightarrow \infty} c_{n}&=\lim _{n \rightarrow \infty} \sqrt{n^{2}+n}-\sqrt{n^{2}-n} \frac{\sqrt{n^{2}+n}+\sqrt{n^{2}-n}}{\sqrt{n^{2}+n}+\sqrt{n^{2}-n}}\\ &= \lim _{n \rightarrow \infty} \frac{2 n}{\sqrt{n^{2}+n}+\sqrt{n^{2}-n}}\\ &=\lim _{n \rightarrow \infty} \frac{2 }{\sqrt{1+1/n}+\sqrt{1-1/n}}\\ &=1 \end{align*}