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Unformatted text preview: d dx of (1 + x ) n ?) Problem 4. Note that we can write ± n k ² = n ! k !( nk )! = ( n ) k k ! , where ( n ) k := n ( n1) ··· ( n( k1)). Why would we do this? Because the expression ( z ) k makes sense for any positive integer k and any complex number z ∈ C . Thus we can deﬁne ( z k ) := ( z ) k /k ! for any k ∈ N and z ∈ C . Prove that for all n,k ∈ N we have ±n k ² = (1) k ± n + k1 k ² . Problem 5. Let x,z ∈ C be complex numbers with  x  < 1. Newton’s Generalized Binomial Theorem says that (1 + x ) z = 1 + ± z 1 ² x + ± z 2 ² x 2 + ± z 3 ² x 3 + ··· where the right hand side is a convergent inﬁnite series. Use this to obtain an inﬁnite series expansion of (1 + x )2 when  x  < 1. (Hint: Apply Problem 4.)...
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This note was uploaded on 01/08/2012 for the course MATH 461 taught by Professor Armstrong during the Fall '11 term at University of Miami.
 Fall '11
 Armstrong
 Math, Binomial Theorem, Binomial

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