# How do you determine whether the sequence a_n=sqrtn converges, if so how do you find the limit?

Apr 25, 2017

${\lim}_{n \to \infty} \sqrt{n} = \infty$

#### Explanation:

This is quite intuitive, as when $n$ grows, so does its square root, without bounds.

We can formally demonstrate it using the definition of the limit: given any $M > 0$, if we choose $N > {M}^{2}$ we have for $n > N$:

${a}_{n} = \sqrt{n} > \sqrt{N} > \sqrt{{M}^{2}}$

so that:

$n > N \implies {a}_{n} > M$

which proves that:

${\lim}_{n \to \infty} \sqrt{n} = \infty$