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Essentials of Mathematical Probability and Statistics

John Travis

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Section 12.2 Binomial SumsBinomial SeriesTrinomial Series

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The binomial series is also foundational. It is technically not a series since the sum_if finite but we won’t bother with that for now. It is given by
B=∑k=0n(nk)akbn−k
By induction:
Basic Step: n = 1 is trivial
Inductive Step: Assume the statement is true as given for some n≥1. Show (a+b)n+1=∑k=0n+1(n+1k)akbn+1−k
(a+b)n+1=(a+b)(a+b)n=(a+b)∑k=0n(nk)akbn−k=∑k=0n(nk)ak+1bn−k+∑k=0n(nk)akbn−k+1=∑k=0n−1(nk)ak+1bn−k+an+1+bn+1+∑k=1n(nk)akbn−k+1=∑j=1n(nj−1)ajbn−(j−1)+an+1+bn+1+∑k=1n(nk)akbn+1−k=bn+1+∑k=1n[(nk−1)+(nk)]akbn+1−k+an+1=bn+1+∑k=1n(n+1k)akbn+1−k+an+1=∑k=0n+1(n+1k)akbn+1−k
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Consider B(a,b)=∑k=0n(nk)akbn−k. This finite sum_is known as the Binomial Series.
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Show that B(a,b)=(a+b)n
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Show that B(1,1)=2n
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Show that B(−1,1)=0
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Show that B(p,1−p)=1
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Easily, B(x,1)=∑k=0n(nk)ak
(a+b+c)n=∑k1+k2+k3=n(nk1,k2,k3)ak1bk2ck3
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where (nk1,k2,k3)=n!k1!k2!k3!. This can be generalized to any number of terms to give what is know as a multinomial series.
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