I Proving the Finite Binomial Series for k Non-Negative Integer

Adrenaline101
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What is the method to prove that a binomial series is not infinite when k is a non-negative integer.
Hello,

I was wondering how to prove that the Binomial Series is not infinite when k is a non-negative integer. I really don't understand how we can prove this. Do you have any examples that can show that there is a finite number when k is a non-negative integer?

Thank you!
 
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Do you mean the series expansion of ##(1+x)^k##? Why should it be infinite?
 
Gaussian97 said:
Do you mean the series expansion of ##(1+x)^k##? Why should it be infinite?
Yes, I mean that expansion. I was wondering how to prove that this series expansion is finite when k i a non-negative integer, because eventually, if the value of k is non-negative, this binomial series's values will eventually be 0.
I just cannot find a written proof that allows me to understand why this is happening.
 
Shouldn't the fact that it is a finite degree polynomial be enough proof?
 
Adrenaline101 said:
Yes, I mean that expansion. I was wondering how to prove that this series expansion is finite when k i a non-negative integer, because eventually, if the value of k is non-negative, this binomial series's values will eventually be 0.
I just cannot find a written proof that allows me to understand why this is happening.
Any real analysis textbook should cover the convergence of power series. In general, a power series has a radius of convergence: e.g. for the binomial expansion:
$$(1 - x)^{-1} = 1 + x + x^2 + x^3 \dots$$ converges for ##|x| < 1## and diverges otherwise.
 
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Adrenaline101 said:
Summary:: What is the method to prove that a binomial series is not infinite when k is a non-negative integer.

Hello,

I was wondering how to prove that the Binomial Series is not infinite when k is a non-negative integer. I really don't understand how we can prove this. Do you have any examples that can show that there is a finite number when k is a non-negative integer?

Thank you!
You mean ##k## is a negative integer, right? When ##k## is a positive integer, the expansion has a finite number of terms.
 
Oh I see... Sorry, I was stuck on a particular idea, which is more simple in reality.
Thank you!
 
Let ##n ## be a non-negative integer. If ##n=0## then for ## a \neq 0, a^n=1##, If not, :

## (1 +x )^n = (1+x)(1+x)...(1+x) ## (n times) . The expression consists of a total of n expressions,each with two terms. If you want more rigor, do induction on n. If the expression ##(1+x(^k## is finite, so is ##(1+x)^{k+1} =(1+x)^k (1+x)## . If not, consider that the expanded expression consists of all monomials ## a_n x^n ## given by ## \Sigma_{i=0}^n nCi x^i ##, where nCi is "n Choose i" . This is a finite sum with (n+1) terms.
 
There's various way to do it:

(1) (1 + x)^k is a product of k polynomials of degree 1, and is therefore a polynomial of degree k.

(2) We have (1 + x)^\alpha = \sum_{n=0}^\infty a_n x^n where a_{n+1} = \frac{(\alpha - n)}{n+1}a_{n} with a_0 = 1. If \alpha is a non-negative integer then a_{\alpha + 1} = \frac{(\alpha - \alpha)}{\alpha + 1}a_\alpha = 0 and thus a_n = 0 for all n &gt; \alpha.
 
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A compact way: <br /> 2^{n}=(1+1)^{n}=\int_{k=0}^{n}\begin{pmatrix}<br /> k\\<br /> n\\<br /> \end{pmatrix}1^{k}\cdot 1^{n-k}=\int_{k=0}^{n}\begin{pmatrix}<br /> k\\<br /> n\\<br /> \end{pmatrix}<br />
 
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