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

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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

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

Im having trouble finding the limit

##### 2 Answers

#### Explanation:

You are on the good path:

Now as:

we have:

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.

(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

Then, *lesser* denominator.

We should see that

Then, by the direct comparison test,