# How do you determine whether the sequence #a_n=(n+1)^n/n^(n+1)# converges, if so how do you find the limit?

Note that:

So:

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To determine whether the sequence ( a_n = \frac{(n+1)^n}{n^{n+1}} ) converges, we can investigate its behavior as ( n ) approaches infinity.

We can rewrite ( a_n ) as:

[ a_n = \frac{(n+1)^n}{n^{n+1}} = \frac{(1 + \frac{1}{n})^n}{n} ]

The sequence ( a_n ) converges if the limit of ( a_n ) as ( n ) approaches infinity exists.

We can use the fact that ( \lim_{n \to \infty} (1 + \frac{1}{n})^n = e ).

Thus,

[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{(1 + \frac{1}{n})^n}{n} = \frac{e}{\infty} = 0 ]

Since the limit of ( a_n ) as ( n ) approaches infinity is 0, the sequence ( a_n ) converges.

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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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