a. b. c. d. e. a. b. c. d. e. Solution binomial series. can be solved with the f

Question

a. b. c. d. e. a. b. c. d. e.

Explanation / Answer

binomial series. can be solved with the formula .................8..[ k] ( 1 + x)^k = ? [ ] x^n ..............n=0.[n ] our function is [ 1 / (x+1)^4 ] =====> can be expressed as (1 + x )^-4 8....[ - 4 ] ?.. [ ] (x)^n n=0..[ n ] where [ -4] [ ] =====> [ -4! / ( n! * ( -4 - n )! ] [n ] 8 ? [ -4! / ( n! * ( -4 - n )! ] * (x)^n n=0 ratio test : an+1 / an [ -4! / ( (n+1)! * ( -4 - (n+1) )! ] * (x)^(n+1) / [ -4! / ( n! * ( -4 - n )! ] * (x)^n [ -4! / ( (n+1) * n! * ( -4 - n-1) )! ] * (x)^(n) * x^1) * [ ( n! * ( -4 - n )! / -4! (x)^n ] [ ( -4 - n )! * x / ( (n+1) * ( -5 - n)! ] ) [ ( -4 - n ) ( -5 - n )! * x / ( (n+1) * ( -5 - n)! ] ) ( -4 - n ) x / ( (n+1) ) lim x * [ ( -4 - n ) / (n+1) ] ====> large in charge n-->8 lim x * [ -n / n ] n-->8 lim x * -1 = -x ====> I x I < 1 n-->8 the radius of convergence R = 1

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