Find a power series representation for the function and determine the radius of

Question

Find a power series representation for the function and determine the radius of convergence. a) f(x) = 1/(4+x)^2 b) f(x) = 1/(4+x)^3 c) f(x) = (x^2)/(4+x)^3

Explanation / Answer

A ) this is called the binomial series. can be solved with the formula .................8.. .[ k] ( 1 + x)^k = ? .[ ] x^n = [ 1 + kx + ( k (k - 1) / 2! )x^2 + ( k (k - 1)(k - 2) / 3! )x^3 + ....] ..............n=0. .[n ] our function is (x/2-x)^3 =====> can be expressed as [ x^3 / (2-x)^3 ] [ x^3 / ( 2(1- (x/2) ) )^3 ] x^3 * [ 1 / 8(1- (x/2) )^3 ] (x^3/8) * [ 1 / (1+ (-x/2) )^3 ] (x^3/8) * (1+ (-x/2) )^-3 8....[(-3)] ?.. [ ] (-x/2)^n * (x^3/8) n=0..[ n ] 8....[(-3)] ?.. [ ] [ (-1)^n * (x)^n * x^3 ] / [ 2^n * 8) ] n=0..[ n ] 8....[(-3)] ?.. [ ] [ (-1)^n * (x)^(n+3) ] / [ 2^n * 2^3) ] n=0..[ n ] 8....[(-3)] ?.. [ ] [ (-1)^n * (x)^(n+3) ] / [ 2^(n+3) ] n=0..[ n ] 8 ? [ (-3)! (-1)^n * (x)^(n+3) ] / [ 2^(n+3) * n! ] n=0 [ ( (-3) * (-4) * (-5) * (-6) * (-7)...) (-1)^n * (x)^(n+3) ] / [ 2^(n+3) * (1 * 2 * 3 * 4 * 4 * 5 * 6....) ] [ (-1)^n( (3) * (4) * (5) * (6) * (7)...) (-1)^n * (x)^(n+3) ] / [ 2^(n+3) * (1 * 2 * 3 * 4 * 4 * 5 * 6....) ] [ (1)^n( (3) * (4) * (5) * (6) * (7)...) * (x)^(n+3) ] / [ 2^(n+3) * (1 * 2 * 3 * 4 * 4 * 5 * 6....) ] [ (1)^n * (x)^(n+3) ] / [ 2^(n+3) * (1 * 2) ] [ (1)^n * (x)^(n+3) ] / [ 2^(n+3) * 2 ] [ (1)^n * (x)^(n+3) ] / [ 2^(n+3+1) ] [ (1)^n * (x)^(n+3) ] / [ 2^(n+4) ] using ration test: (an + 1) / (an) [ (1)^(n+1) * (x)^(n+3+1) ] / [ 2^(n+4+1) ] ÷ [ (1)^n * (x)^(n+3) ] / [ 2^(n+4) ] [ (1)^(n) (1)^1 * (x)^(n+4) ] / [ 2^(n+5)! ] * [ 2^(n+4) ] / [ (1)^n * (x)^(n+3) ] [ (x)^(n) x^(4) ] / [ 2^(n) 2^(5) ] * [ 2^(n) 2^(4) ] / [ (x)^(n) x^(3) ] [ x ] / [ 2 ] lim (- x/2) = (- x/2) = I (- x/2) I < 1 = x/2 < 1 ===> x < 2 n-->8 Radius of Convergence is 2

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