Chapter 2.5, Problem 90E

Looking ahead to sequences A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6, 8, … } is specified by the function f(n) = 2n, where n = 1, 2, 3, …. The limit of such a sequence is $\underset{n\to \infty }{\lim }f(n)$, provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist.

78. $\left\{\frac{1}{2},\frac{4}{3},\frac{9}{4},\frac{16}{5},\dots \right\}$, which is defined by $f(n)=\frac{n^2}{n+1}$, for n = 1, 2, 3, …