How do you determine if the sum of #5^n/(3^n + 4^n)# from n=0 to infinity converges?
That sum diverges.
Since it is a sum of all positive numbers, it is regular (convergent or divergent, not irregular).
Since:
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To determine if the series (\sum_{n=0}^\infty \frac{5^n}{3^n + 4^n}) converges, you can use the ratio test.

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} ).

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 ]

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.

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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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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