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

1 Answer
May 18, 2017

By putting an limit in front.

Explanation:

lim_(n=>oo)((3n-7)/(10n+9))
In this case, if the result is !=0 there's divergence.
But the result is something that we cannot solve.
=>[oo/oo]
So we are going to isolate n in the equation:
=>lim_(n=>oo)((n(3-7/n))/(n(10+9/n)))
=lim_(n=>oo)((3-7/n)/(10+9/n))
And we just have to solve it!
=(3-7/oo)/(10+9/oo)
=(3-0)/(10+0)
=3/10
The result is !=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.)