How do you use the ratio test to test the convergence of the series #∑(n!)/(n^n)# from n=1 to infinity?
The series converges.
The term of the series is
The ratio test is
Therefore,
As
The series converges.
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To use the ratio test to test the convergence of the series (\sum \frac{n!}{n^n}) from (n=1) to infinity, follow these steps:

Calculate the limit of the ratio (L) as (n) approaches infinity: [L = \lim_{n \to \infty} \left\frac{a_{n+1}}{a_n}\right] where (a_n = \frac{n!}{n^n}).

Compute the ratio of consecutive terms: [\frac{a_{n+1}}{a_n} = \frac{(n+1)!}{(n+1)^{n+1}} \cdot \frac{n^n}{n!}] [= \frac{(n+1)!}{n!} \cdot \frac{n^n}{(n+1)^{n+1}}] [= \frac{n+1}{(1+\frac{1}{n})^n}]

Take the limit as (n) approaches infinity: [L = \lim_{n \to \infty} \frac{n+1}{(1+\frac{1}{n})^n}]

Determine the convergence of the series based on the value of (L):
 If (L < 1), the series converges absolutely.
 If (L > 1) or (L = \infty), the series diverges.
 If (L = 1), the ratio test is inconclusive, and other tests may be needed to determine convergence.

Therefore, to test the convergence of the series (\sum \frac{n!}{n^n}), calculate the limit (L) and compare it to 1 to make the determination.
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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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