# #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

To determine convergence or divergence, we'll use the ratio test.

Let ( a_n = \frac{5^n}{3^n + 4^n} ).

Calculate the limit of the ratio: [ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{5^{n+1}}{3^{n+1} + 4^{n+1}} \cdot \frac{3^n + 4^n}{5^n} \right| ]

[ = \lim_{n \to \infty} \left| \frac{5}{3 + \left(\frac{4}{3}\right)^n} \right| = \frac{5}{3} ]

Since the limit is less than 1, by the ratio test, the series converges.

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(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.)

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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.

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