Question

Proof: \(\frac{n^{2}+5}{n+1} \rightarrow \infty\) as \(n \rightarrow \infty\)

Answer 1

Proof. Let \(a_{n}:=\frac{n^{2}+5}{n+1}\) and let \(A>0\) be given. Observe that

\(\begin{aligned} a_{n}:=\frac{n^{2}+5}{n+1} & \geq \frac{n^{2}}{n+1} \\ & \geq \frac{n^{2}}{n+n} \\ &=\frac{n^{2}}{2 n} \\ &=\frac{n}{2} \end{aligned}\)

and \(\frac{n}{2}>A\) provided that \(n>2 A\).
So, let \(N\) be any natural number larger that \(2 A\). Then if \(n>N\), we have \(a_{n}>\frac{n}{2}>\frac{N}{2}>A\). Therefore, \(a_{n}\) tends to infinity.