Limit Test on Series: Summation from n=0 to Infinity of n!/1000^n

In summary, the conversation discusses the summation from n=0 to infinity of n!/1000^n and the possibility of using the ratio test to evaluate it. The question of whether n!>1000^n for large n is brought up as a possible approach.
  • #1
cue928
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Series: summation from n=0 to infinity of n!/1000^n

I can look at that and see the limit is not going to zero but how do you show that? Also, were it not in the first section of the book (i.e. before the ratio test), I would have tried to use the ratio test on it - is that acceptable to do? I got infinity for the answer under the ratio test, btw.
 
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  • #2
cue928 said:
Also, were it not in the first section of the book (i.e. before the ratio test), I would have tried to use the ratio test on it - is that acceptable to do?
Sure, why not?

I can look at that and see the limit is not going to zero but how do you show that?
Do you think [itex]n!>1000^{n}[/itex] for some large n to be true? If so, could you manipulate the rational function using this knowledge?
 
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FAQ: Limit Test on Series: Summation from n=0 to Infinity of n!/1000^n

1. What is a limit test on a series?

A limit test on a series is a method used to determine if the terms of a series approach a limit (a specific value or infinity) as the number of terms becomes infinite. It is used to determine whether a series converges (approaches a finite value) or diverges (does not approach a finite value).

2. What does the series n!/1000^n represent?

The series n!/1000^n represents the sum of the terms from n=0 to infinity of the factorial of n divided by 1000 to the power of n. This can also be written as 1/1000 + 1/1000^2 + 1/1000^3 + ...

3. How is a limit test on a series performed?

A limit test on a series is typically performed by evaluating the limit of the nth term of the series as n approaches infinity. This can be done using various methods, such as the comparison test, the ratio test, or the root test.

4. What is the significance of testing a series for convergence or divergence?

Testing a series for convergence or divergence is important because it helps us determine whether the sum of the terms in the series will approach a finite value or not. This can have important implications in various areas of mathematics, such as calculus and probability theory.

5. What is the application of the limit test on the series n!/1000^n?

The limit test on the series n!/1000^n can be applied in various areas of mathematics, such as in the analysis of algorithms, probability theory, and in the study of exponential growth and decay. It can also be used to evaluate infinite series and to determine the convergence or divergence of certain mathematical expressions.

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