How do you determine whether the sequence #a_n=(n!+2)/((n+1)!+1)# converges, if so how do you find the limit?

Answer 1

The sequence #a_n# converges to #0#. The series (i.e. sum) diverges.

Let:

#a_n = (n!+2)/((n+1)!+1)#
#color(white)(a_n) = (n!+2)/((n+1)n!+1)#
#color(white)(a_n) = (1+2/(n!))/(n+1+1/(n!))#
As #n->oo#:
#2/(n!)->0#
#1/(n!)->0#

So:

#a_n->(1+0)/(oo+1+0) = 0#
So the sequence #a_1, a_2, a_3,...# tends to #0# as #n->oo#

How about the sum?

#a_n = (n!+2)/((n+1)!+1) > (n!)/((n+1)!+1) = 1/(n+1+1/(n!)) >= 1/(n+2)#

So:

#sum_(n=1)^oo a_n >= sum_(n=1)^oo 1/(n+2) = sum_(n=3)^oo 1/n#
which diverges, since the harmonic series #sum_(n=1)^oo 1/n# diverges.
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Answer 2

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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Answer from HIX Tutor

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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