How do you use the ratio test to test the convergence of the series #∑ (8^n)/(n!)# from n=1 to infinity?
The series would converge.
To test the convergence apply the ratio test which means to find
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To use the ratio test to test the convergence of the series (\sum \frac{8^n}{n!}) from (n=1) to infinity, follow these steps:

Compute the ratio of consecutive terms of the series: [ \lim_{n \to \infty} \left \frac{a_{n+1}}{a_n} \right = \lim_{n \to \infty} \left \frac{\frac{8^{n+1}}{(n+1)!}}{\frac{8^n}{n!}} \right ]

Simplify the ratio: [ = \lim_{n \to \infty} \left \frac{8^{n+1} \cdot n!}{8^n \cdot (n+1)!} \right ]

Cancel common terms: [ = \lim_{n \to \infty} \left \frac{8}{n+1} \right ]

Evaluate the limit: [ = 0 ]

Analyze the limit:
 If the limit is less than 1, then the series converges absolutely.
 If the limit is greater than 1 or infinite, then the series diverges.
 If the limit equals 1, the test is inconclusive.
Since the limit is 0, which is less than 1, by the ratio test, the series (\sum \frac{8^n}{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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