How do you use the Ratio Test on the series #sum_(n=1)^oo(n!)/(100^n)# ?

Answer 1

#limnrarroo ((n+1)!)/(100^(n+1))/(n!)/100^n#

#limnrarroo ((n+1)!)/(100^(n+1))/(n!)/100^n#
#limnrarroo ((n+1)!)/(100^(n+1))*(100^n)/(n!)#
#limnrarroo ((n+1))/(100^(n+1))*(100^n)#
#limnrarroo (n+1)/(100)#
#=oo/100=oo#

The Ratio Test states that if this limit is greater than 1, the series diverges. Less than 1, it converges. If it is exactly 1, the test is inconclusive.

Because #oo# is obviously greater than 1, the series diverges.
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Answer 2

To use the Ratio Test on the series ( \sum_{n=1}^{\infty} \frac{n!}{100^n} ), we compute the limit:

[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| ]

where ( a_n = \frac{n!}{100^n} ).

[ \lim_{n \to \infty} \left| \frac{(n+1)!/100^{n+1}}{n!/100^n} \right| = \lim_{n \to \infty} \left| \frac{(n+1)!}{n!} \times \frac{100^n}{100^{n+1}} \right| ]

[ = \lim_{n \to \infty} \left| \frac{n+1}{100} \right| = \frac{1}{100} ]

Since the limit is less than 1, the Ratio Test tells us that the series converges.

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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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