Question

`sum_(n=0)^oo 5^n/(3^n +4^n)`. Does the series converge or diverge?

I used Divergence Test and I'm stuck on this part

`lim_(n->oo) (5/4)^n/[(3/4)^n +1]`

Im having trouble finding the limit

Answer 1

`sum_(n=0)^oo 5^n/(3^n+4^n) = +oo`

Explanation:

You are on the good path:

`5^n/(3^n+4^n) = (5/4)^n/(1+(3/4)^n)`

Now as:

`lim_(n->oo) 1/(1+(3/4)^n) = 1`

we have:

`lim_(n->oo) (5/4)^n/(1+(3/4)^n) = lim_(n->oo) (5/4)^n * lim_(n->oo) 1/(1+(3/4)^n) = +oo * 1 = +oo`

and as the sequence of the terms is not infinitesimal, the series cannot converge. As the terms are all positive the series is then divergent.

Answer 2

(I know this isn't the method you requested, but this is how I first approached the problem. With series problems, there are frequently many valid solutions.)

Explanation:

Note that `3^n+4^n<4^n+4^n=2(4^n)`.

Then, `5^n/(3^n+4^n)>5^n/(2(4^n))` since the first one has a lesser denominator.

We should see that `sum_(n=0)^oo5^n/(2(4^n))=1/2sum_(n=0)^oo(5/4)^n`, which is divergent by the Geometric Series test since divergent since `abs(5/4)>1`.

Then, by the direct comparison test, `sum_(n=0)^oo5^n/(3^n+4^n)` is divergent as well since it is larger than another divergent series.