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QUESTION
The Binomial Theorem can be extended to include complex numbers. (a) Prove that for all real numbers and non-negative integers n, Sum n k=0 C(n k)...
The Binomial Theorem can be extended to include complex numbers.
(a) Prove that for all real numbers θ and non-negative integers n, Sum n k=0 C(n k) (cos kθ + i*sin kθ) = (1 + cos θ + i*sinθ )^n .
(b) Using part (a), prove that for all real numbers θ and non-negative integers n, Sum n k=0 C(n k) (cos kθ) = (2 cos θ/2)^n * cos nθ/2 n sin nθ 2 .
