How do you apply the ratio test to determine if $sum_(n=1)^oo 3^n$ is convergent to divergent?

Question

2096 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-01-15T03:05:06

The series diverges.

The the $n$ th term in the series is given by $a_n=3^n.$

The ratio test states that the series should be convergent if:

$lim_(n->oo)a^(n+1)/a^n<1$

So, in our case we have:

$lim_(n->oo)3^(n+1)/3^n=lim_(n->oo)3^(n+1-n)=$

$=lim_(n->oo)3=3$

$3gt1$ so the ratio test tells us this series diverges.

How do you apply the ratio test to determine if sum_(n=1)^oo 3^nsum_(n=1)^oo 3^n is convergent to divergent?