BinomialTheorem - The Binomial Theorem Notation 1 Dene 0 = 1 and recursively(n 1 = n(n 1 For 0 k n dene n k = n k(n k Lemma 0.1 If 0 j n then n n j1 j n

# BinomialTheorem - The Binomial Theorem Notation 1 Dene 0 =...

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The Binomial Theorem Notation 1. Define 0! = 1 and, recursively, ( n + 1)! = n !( n + 1). For 0 k n define n k = n ! k !( n - k )! Lemma 0.1. If 0 j n then n j - 1 + n j = n + 1 j Proof. Exercise. Theorem 1 (Binomial Theorem) . For any real numbers a and b and any n N ( a + b ) n = n X j =0 n j a j b n - j Proof. Proceed by induction on n noting that the case n = 1 is easy. Given the theorem for n it must be shown that ( a + b ) n +1 = n +1 X j =0 n + 1 j a j b n +1 - j . To this end, note that ( a + b ) n +1 = ( a + b n ( a + b ) = n X j =0 n j a j b n - j ( a + b ) = n X j =0 n j a j +1 b n - j + n X j =0 n j a j b n +1 - j = = n +1 X j =1 n j