How do you determine whether the sequence #a_n=(n!+2)/((n+1)!+1)# converges, if so how do you find the limit?
The sequence
Let:
So:
How about the sum?
So:
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To determine convergence of the sequence (a_n = \frac{{n! + 2}}{{(n + 1)! + 1}}), we find its limit as (n) approaches infinity.
[ \lim_{{n \to \infty}} a_n = \lim_{{n \to \infty}} \frac{{n! + 2}}{{(n + 1)! + 1}} ]
To find the limit, we can simplify the expression:
[ \lim_{{n \to \infty}} a_n = \lim_{{n \to \infty}} \frac{{n! + 2}}{{(n + 1)! + 1}} = \lim_{{n \to \infty}} \frac{{\frac{{n!}}{{(n + 1)!}} + \frac{2}{{(n + 1)!}}}}{{1 + \frac{1}{{(n + 1)!}}}} ]
Since (n!/(n + 1)! = 1/(n + 1)) and (2/(n + 1)!) and (1/(n + 1)!) tend to zero as (n) goes to infinity, we get:
[ \lim_{{n \to \infty}} a_n = \frac{1}{1} = 1 ]
Therefore, the sequence converges to (1).
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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.
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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