Factorials within alternating series

In summary, the problem involves finding the limit of the series ∑ [ (-1)^n * n!/(10^n) ] and determining if it converges or diverges. The difficulty is in using derivatives or L'Hospital's rule to prove that the nth term of the series goes to zero. However, it is possible to show that the series diverges by noting that the terms are positive and increasing for large values of n.
  • #1
ahmed markhoos
49
2

Homework Statement



∑ [ (-1)^n * n!/(10^n) ]

2. The attempt at a solution

the problem is that I cannot use derivative to make sure that a(n) is decreasing neither L hopital rule to find the limit.
 
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  • #2
ahmed markhoos said:

Homework Statement



∑ [ (-1)^n * n!/(10^n) ]

2. The attempt at a solution

the problem is that I cannot use derivative to make sure that a(n) is decreasing neither L hopital rule to find the limit.

Have you thought about whether the nth term of the series goes to zero?
 
  • #3
LCKurtz said:
Have you thought about whether the nth term of the series goes to zero?
I don't know if this is correct or not! because I've used L' hopital rule for one side and left the side of n! without derivation

##\lim_{n\rightarrow \infty} {\frac{n!}{10^n}}##

## ln(f(n)) = ln{\frac {n!}{10^n}} ##
##ln(f(n)) = ln(n!) - ln(10^n)##
##ln(f(n)) = ln(n!) - n*ln(10)##

##\lim_{n\rightarrow \infty} {ln(n!) - n*ln(10)}##

Using L' hopital rule ##\lim_{n\rightarrow \infty} {ln(n!) - ln(10)}\ = ∞ ##

since: ##ln(f(n)) = ∞## $$f(n) = e^∞ = ∞$$

which make the series diverges, is this correct ?
 
  • #4
That isn't the form for L'Hospital's rule and you certainly don't need L'Hospital's rule for this problem. Why don't you just note the terms are positive and check that they are increasing for large ##n##?
 
Last edited:

1. What are factorials within alternating series?

Factorials within alternating series refer to a mathematical concept where the terms in a series alternate between being multiplied by a factorial and being divided by a factorial. This results in a series that does not converge to a single value, but rather oscillates between two values.

2. How do factorials within alternating series behave?

The behavior of factorials within alternating series depends on the specific series being evaluated. In some cases, the series may converge to a finite value, while in others it may diverge or oscillate between two values. The behavior can be determined by applying mathematical techniques, such as the ratio test or the alternating series test.

3. What is the significance of factorials within alternating series in mathematics?

Factorials within alternating series are important in mathematics for their applications in various fields, such as calculus, number theory, and probability. They also provide insights into the behavior of infinite series and help in determining the convergence or divergence of these series.

4. Can factorials within alternating series be simplified?

In some cases, factorials within alternating series can be simplified by using mathematical techniques such as partial fraction decomposition or the binomial theorem. However, in general, these series cannot be simplified to a closed form and must be evaluated numerically.

5. Are there any real-world applications of factorials within alternating series?

Yes, factorials within alternating series have various real-world applications, such as in calculating probabilities in gambling games, determining the convergence of infinite series in physics and engineering, and in computing error bounds in numerical analysis.

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