(9) Using the Binomial Expansion, determine the coefficient of x-2 in: (.22 +, r 2)2n+1
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
What is the coefficient of x' in the binomial expansion of (3x +5)*?
9. Using the Binomial Theorem, show that Σk㈡-n 2n-1
40. Use the Binomial Theorem to find the 3rd coefficient in the expansion of (x – 3)?.
Problem 7: 15 points. For any a ER, define the binomial-type coefficient by a(a – 1)(a − 2)... (a – n +1) C(a, n) = n! for n > 1 and C(a,0) = 1. For example, C(1,2)= Let g(x) = {nco C(2, n)x". (a) Determine the radius of convergence r for the power series g(x). (b) Determine the power series for the derivative g(x) on the interval (-r,r). (c) Show : g(x) = 2-0 C(,n)z” by examining the coefficients in...
12. Use the binomial theorem to find the coefficient of xayh in the expansion of (5x2 +2y3)6, where a) a 6, b-9 b) a 2, b 15. c) a 3, b 12. d) a 12, b 0 e) a 8, b 9
PLEASE DO BOTH (3) (5 pts) Find the expansion of (2x + y) using the binomial theorem? (4) (5 pts) What is the coefficient of z' in the expansion of (2 + x)?
(b) What is the radius of convergence of the Binomial Series expansion for (x + 1)a? (c) Use the Binomial Series to find the first four nonzero terms of the expansion of (x+1)2/3. (d) Use part (c) to approximate 1.022/3. (b) What is the radius of convergence of the Binomial Series expansion for (x + 1)a? (c) Use the Binomial Series to find the first four nonzero terms of the expansion of (x+1)2/3. (d) Use part (c) to approximate 1.022/3.
(a) Resolve the expansion 2 ac (x - 1)(x + 2) into partial fractions sums. (b) Hence determine the binomial expansion up to and including the term involving x?.
22 . . 23 . 24 Use the binomial series to expand the function as a power series. 7 (4 + x) 3 Σ Your answer cannot be understood or graded. More Information n = 0 X State the radius of convergence, R. R = 4 Use a Maclaurin series in this table to obtain the Maclaurin series for the given function. f(x) = x cos(4x) D) n = 0 Evaluate the indefinite integral as an infinite series. I conte...