# How do you use the ratio test to test the convergence of the series #∑ 3^n/(4n³+5)# from n=1 to infinity?

The series is divergent, see the explanation.

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To use the ratio test to test the convergence of the series ∑ 3^n/(4n³+5), you would take the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term:

lim (n→∞) |(a_{n+1}/a_n)|

In this case, the series is:

a_n = 3^n / (4n³ + 5)

So, you would calculate:

lim (n→∞) |(3^(n+1) / (4(n+1)³ + 5)) / (3^n / (4n³ + 5))|

Simplify the expression and then calculate the limit. If the limit is less than 1, the series converges. If it's greater than 1 or if it doesn't exist, 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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