# How do you test the series #Sigma 5^n/(3^n+4^n)# from n is #[0,oo)# for convergence?

The series is divergent.

so

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You can use the ratio test to test the convergence of the series ( \sum \frac{5^n}{3^n + 4^n} ) as ( n ) approaches infinity.

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} ).

[ \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 \cdot 5^n(3^n + 4^n)}{5^n(3 \cdot 3^n + 4 \cdot 4^n)} \right| ]

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

[ = \lim_{{n \to \infty}} \left| \frac{5(1 + (\frac{4}{3})^n)}{3 + (\frac{4}{3})^n} \right| ]

As ( n ) approaches infinity, ( (\frac{4}{3})^n ) approaches infinity, and the ratio simplifies to:

[ = \lim_{{n \to \infty}} \left| \frac{5}{1} \right| ]

[ = 5 ]

Since the limit is greater than 1, the series diverges.

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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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