How do you determine if the sum of #5^n/(3^n + 4^n)# from n=0 to infinity converges?

Answer 1

That sum diverges.

Since it is a sum of all positive numbers, it is regular (convergent or divergent, not irregular).

Since:

#lim_(nrarr+oo)a_n=lim_(nrarr+oo)5^n/(3^n+4^n)=#
#lim_(nrarr+oo)5^n/4^n=lim_(nrarr+oo)(5/4)^n=+oo#
(#3^n# is negligible respect #4^n#)
and it is not #0#, then it is divegent.
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Answer 2

To determine if the series (\sum_{n=0}^\infty \frac{5^n}{3^n + 4^n}) converges, you can use the ratio test.

  1. Apply the ratio test: [ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| ] where ( a_n = \frac{5^n}{3^n + 4^n} ).

  2. Calculate the limit: [ \lim_{n \to \infty} \left| \frac{\frac{5^{n+1}}{3^{n+1} + 4^{n+1}}}{\frac{5^n}{3^n + 4^n}} \right| ] [ = \lim_{n \to \infty} \left| \frac{5^{n+1}(3^n + 4^n)}{5^n(3^{n+1} + 4^{n+1})} \right| ] [ = \lim_{n \to \infty} \left| \frac{5}{\left( \frac{5}{3} \right)^n \cdot \left( \frac{4}{5} + \left( \frac{4}{3} \right)^n \right)} \right| ]

  3. Determine the value of the limit: Since both ( \left( \frac{5}{3} \right)^n ) and ( \left( \frac{4}{3} \right)^n ) grow without bound as ( n ) approaches infinity, the denominator grows faster than the numerator. Thus, the limit is 0.

  4. Conclusion: Since the limit is less than 1, by the ratio test, the series (\sum_{n=0}^\infty \frac{5^n}{3^n + 4^n}) converges absolutely.

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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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