Using the Binomial Theorem, show that 9. Using the Binomial Theorem, show that Σ
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.
Prove that P2n(0)= (-1)n ((2n-1)!!/(2n)!!) using the generation function and a binomial expansion. Show that (sqrt(pi)(4n-1)/(2gamma(n+1)gamma(3/2-n))=(-1)n-1((2n-3)!!/(2n-2)!!)(4n-1)/2n
(9) Using the Binomial Expansion, determine the coefficient of x-2 in: (.22 +, r 2)2n+1
Using Taylor’s Theorem (and taking x0= 0), show that (for |x| << 1) (1+ x)n ≈ 1+ nx This can be especially useful for approximating the values of square roots, for which n = ½. (The full expansion of (1+x) n is sometimes called a binomial series, and the first order approximation a “binomial approximation.”)
Use the Binomial Theorem to show that Σ(-1): c(n, k)= 0 -0
Proble A Consider t: foMarkower chainXk 0,1,2, where Xk+1, given Xk = n, is Binomial(2n,1/2). Show that the function f(z) = z s harmonic for this Markov chain. T. IS 311101111 Proble A Consider t: foMarkower chainXk 0,1,2, where Xk+1, given Xk = n, is Binomial(2n,1/2). Show that the function f(z) = z s harmonic for this Markov chain. T. IS 311101111
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]
1. Without using the Binomial Theorem, prove that for all non-negative integers n ΣΘ = 2". IM-
3. Use the Division Algorithm (Theorem 6.1.1) to prove that for all n ez+ 6 I n(n +1) (2n +1). 3. Use the Division Algorithm (Theorem 6.1.1) to prove that for all n ez+ 6 I n(n +1) (2n +1).