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Use the Binomial Theorem to show that Σ(-1): c(n, k)= 0 -0 Show transcribed image text
The binomial theorem states that (a + b)n = Σ (prbn_k. (a) Use the binomial theorem to show that 2k-0 W = 2n. (Hint, 2n= (1 + 1)n.) (b) Expand (a2 + b)4.
b) Using the binomial theorem show that Σ (-1)"/2 (n) cos" k(z) sink(z), Σ (-1)(k-1)/2C) cox"-"(x) sink(z). cos(nx) = sin(nx) = COS k-odd 6 marks]
Using the Binomial Theorem, show that 9. Using the Binomial Theorem, show that Σ
E3. Show that Σ(-1)k( ) = 0 for all positive integers n and k with 0-k-n E4. Show that (t) = Σ ( . 71 に0 k+1 k"d) for all positive integers n and k with 0 ksn
1. Use the binomial theorem to show i) Ek_0 () = 21; ii) _o(-1)* (m.) = 0; and finally that the sum of the ( over odd k equals that over the even k and that both are 21-1. (Hint: for iii) add and subtract the results of i) and ii). For i) and ii) put x and y equal to suitable values in the binomial theorem). (15 points)
The binomial coefficients C(N, k) can be defined recursively as follows: C(N,0)=1, C(N,N) = 1, and, for 0 < k < N, C(N, k) = C(N − 1, k) + C(N − 1, k − 1). Implement the following two methods inside BinomialCoefficients class, one uses recursion and the other one uses dynamic programming. 12. (10 Points) The binomial coefficients C(N, k) can be defined recursively as follows: C(N,0-1, C(N,N) = 1, and, for 0 < k < N, C(N,...
9. Using the Binomial Theorem, show that Σk㈡-n 2n-1
evaluate Σ(1) k=1 n (-1)* k+1 Σ(1). A=0
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
Problem 2 Statement: We know that the binomial series k k(k-1) k1 2)(k -n+1 n=0 converges for l < 1. (a) Use the binomial series to find the Taylor series for f(x)Va centered at z 16. What is the radius 16+ ( 16). of convergence R for your Taylor series? Hint: Problem 2 Statement: We know that the binomial series k k(k-1) k1 2)(k -n+1 n=0 converges for l