#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

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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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.)

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.
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Answer 3

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

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.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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