Question

Use divergent test to determine convergence of the following series?

`sum_(n=0)^oo(3^(n+1)/e^n)`

Answer 1

Diverges.

Explanation:

The divergence test, when applied to a series `suma_n` simply entails taking `lim_(n->oo)a_n`:

If `lim_(n->oo)a_n ne0,` the series `suma_n` diverges.

Note, if `lim_(n->oo)a_n=0,` the series doesn't necessarily converge. It may, but it's not guaranteed.

So,

`lim_(n->oo)3^(n+1)/e^n=lim_(n->oo)(3^n3^1)/e^n=3lim_(n->oo)(3/e)^n=3(oo)=oo`

Thus, the series diverges.