Exploring the Binomial Series: \alpha < n?

In summary, the conversation revolves around the confusion surrounding the existence of the binomial series when alpha is less than n. The experts explain that the series is defined to be zero for n>alpha and there are other ways to define the binomial coefficient for non-integer alpha. The conversation ends with a thank you to Matt for clarifying the confusion.
  • #1
steven187
176
0
hello all

I thought this might be an interesting question to ask, consider the following series
[tex]\sum_{n=0}^{\infty}\left(\begin{array}{cc}\alpha\\n \end{array}\right)x^{n}=(1+x)^{\alpha}[/tex]
this is known as the binomial series, what's confusing me is that how could this series exist when [tex]\alpha< n[/tex] especially when its a series that adds infinitely
number of terms, from my understanding this [tex]\left(\begin{array}{cc}\alpha\\n \end{array}\right)[/tex] can only be evaluated when [tex]\alpha>n[/tex] please help

thanxs
 
Last edited:
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  • #2
they are identically zero for n>alpha (assuming alpha is an integer)
 
  • #3
hello Matt
well I am still confused, what I am not understanding is that if

[tex]\left(\begin{array}{cc}\alpha\\n \end{array}\right)=\frac{\alpha !}{n!(\alpha-n)!}[/tex] and [tex]\alpha< n[/tex] then [tex]\alpha-n[/tex] is negetive how could you find the factorial of a negetive

number, and if they were identically equal to zero i couldn't see how this
[tex]\frac{\alpha !}{n!(\alpha-n)!}=0[/tex]
 
  • #4
They are *DEFINED* to be zero. Who said that that formula holds for the cases you have a problem with?

there is another way to define the binomial coefficient for non-integer alpha too, it's


a(a-1)(a-2)..(a-n+1)/n!

that is defined for all a and all integer n.
 
  • #5
yeah I get what you mean, its just that I couldn't find any site that tells what happens in that sanario so i got really confused anyway

Thanxs Matt
 
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1. What is the binomial series and how does it relate to \alpha < n?

The binomial series is a mathematical series that represents the expansion of a binomial expression. It is used to find the value of a binomial expression for any given value of x. The \alpha < n condition indicates that the series is being explored for values of x that are less than n.

2. Why is it important to explore the binomial series for \alpha < n?

Exploring the binomial series for \alpha < n allows us to understand the behavior of the series for values of x that are smaller than the value of n. This information is useful in various mathematical applications, such as finding approximations for functions and solving problems in probability and statistics.

3. How is the binomial series expanded for \alpha < n?

The binomial series for \alpha < n is expanded using the formula (1+x)^n = 1 + nx + (n(n-1)/2!)x^2 + (n(n-1)(n-2)/3!)x^3 + ..., where n is a positive integer. This formula can be used to find the value of a binomial expression for any value of x that is smaller than n.

4. Are there any specific applications of exploring the binomial series for \alpha < n?

Yes, there are various applications of exploring the binomial series for \alpha < n. One example is in the field of probability, where the series can be used to find the probability of a certain number of successes in a given number of trials. Another application is in physics, where the series can be used to approximate the behavior of a physical system.

5. What are the limitations of exploring the binomial series for \alpha < n?

One limitation is that the series may not converge for values of x that are greater than n. In addition, the series may not accurately represent the behavior of a function for values of x that are significantly smaller than n. It is important to consider the range of values for which the series is being explored and to use other methods if necessary.

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