Convergence or Divergence of Factorial Series

In summary, the conversation discusses the methods used in determining the convergence or divergence of a factorial series. The integral test and divergence test are not applicable, but the ratio test is used to prove that the given factorial series is convergent. However, it is not true that all factorial series are convergent, as shown by the example of \Sigma\frac {n^n}{n!}.
  • #1
Christian M.
2
0
How can I find out if 1/n! is divergent or convergent?

I cannot solve it using integral test because the expression contains a factorial.

I also tried solving it using Divergence test. The limit of 1/n! as n approaches infinity is zero. So it follows that no information can be obtained using this test.

Is there any way that I can prove its divergence or convergence?
 
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  • #2
  • #3
I used the ratio test and got zero as the final answer. So, this means that the given factorial series is convergent.

Just for a follow-up question, is it true then that all factorial series are convergent?
 
  • #4
Christian M. said:
I used the ratio test and got zero as the final answer. So, this means that the given factorial series is convergent.

Just for a follow-up question, is it true then that all factorial series are convergent?
If you are asking about any series summing reciprocals of factorials, the answer is yes as long as they are all different, since any such series is bounded by the sum of all of them (which = e).
 
  • #5
You can use either ratio test or Taylor series for ##e^x ## at ##x=1 ##, as proposed already , or the comparison of ##1/n! ## with ##1/n^2 ##, noticing that for n>3, ##n!>n^2 ##
 
  • #6
Christian M. said:
Just for a follow-up question, is it true then that all factorial series are convergent?
Take a look at [tex]\Sigma\frac {n^n}{n!}[/tex].
 
  • #7
phion said:
Take a look at [tex]\Sigma\frac {n^n}{n!}[/tex].
Obviously not. The sequence [itex]\frac{n^n}{n!}[/itex] itself diverges, so there is no way you could sum it.
 
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Related to Convergence or Divergence of Factorial Series

1. What is the definition of convergence and divergence in relation to factorial series?

Convergence and divergence refer to the behavior of a mathematical series as the number of terms in the series increases. A series is said to converge if the sum of its terms approaches a finite value as the number of terms increases. It is said to diverge if the sum of its terms does not approach a finite value, but instead either increases or decreases without bound.

2. How can I determine if a factorial series converges or diverges?

There are several tests that can be used to determine the convergence or divergence of a factorial series, including the ratio test, the root test, and the integral test. These tests involve comparing the given series to a known convergent or divergent series, or using calculus to evaluate the behavior of the series.

3. What is the significance of convergence or divergence in factorial series?

The convergence or divergence of a factorial series is important in determining the validity and usefulness of the series. A convergent series can be used to approximate values of certain functions, while a divergent series may not have any practical applications.

4. Can a factorial series converge and diverge at the same time?

No, a factorial series can only either converge or diverge. It cannot have both properties simultaneously. However, a series can oscillate between convergence and divergence, which is known as conditional convergence.

5. Are there any real-world applications of convergence or divergence in factorial series?

Yes, factorial series and their convergence or divergence have applications in various fields such as physics, engineering, and economics. In physics, factorial series are used to approximate the values of certain physical phenomena, while in economics, they are used to model growth rates and interest rates.

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