# How do you test for convergence of Sigma (3n-7)/(10n+9) from n=[0,oo)?

May 18, 2017

By putting an limit in front.

#### Explanation:

${\lim}_{n \implies \infty} \left(\frac{3 n - 7}{10 n + 9}\right)$
In this case, if the result is $\ne 0$ there's divergence.
But the result is something that we cannot solve.
$\implies \left[\frac{\infty}{\infty}\right]$
So we are going to isolate n in the equation:
$\implies {\lim}_{n \implies \infty} \left(\frac{n \left(3 - \frac{7}{n}\right)}{n \left(10 + \frac{9}{n}\right)}\right)$
$= {\lim}_{n \implies \infty} \left(\frac{3 - \frac{7}{n}}{10 + \frac{9}{n}}\right)$
And we just have to solve it!
$= \frac{3 - \frac{7}{\infty}}{10 + \frac{9}{\infty}}$
$= \frac{3 - 0}{10 + 0}$
$= \frac{3}{10}$
The result is $\ne 0$ so there's divergence!
(I'not an english speaker so I don't know the name of this method. Feel free to add it.)