How to prove the convergence of the sequence $\lim_{n \rightarrow \infty} \frac{n^2 +1}{2n} $

Question

Using the appropriate definition: $\lim_{n \rightarrow \infty} a_n = + \infty \leftrightarrow \forall_{M\in \mathscr R} \exists_{ n_0\in \mathscr N+} \forall_{ n>n_0}$$ a_n> M$

check that the sequence diverges (that it converges to $+\infty$).

$\lim_{n \rightarrow \infty} \frac{n^2 +1}{2n} $

Answer 1

For all $M \in \mathbb{R}$, let $N=2 \lceil M \rceil$, then for $n>N$, we have $$\frac{n^2+1}{2n} > \frac{n^2}{2n}=\frac{n}{2}>\frac{N}{2}\ge M$$

Answer 2

write your term in the form $$\frac{n^2+1}{2n}=\frac{n}{2}+\frac{1}{2n}$$ this tends to infinity, if $n$ tends to infinity then you must solve the inequality $$\frac{n^2+1}{2n}>M$$