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Anonymous
Anonymous asked in Science & MathematicsMathematics · 8 years ago

Find a power series representation for the function ....?

Find a power series representation for the function

f(x)= x^3 /(1+3*(x^2))^2 and indicate its radius of convergence.

2 Answers

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  • kb
    Lv 7
    8 years ago
    Favorite Answer

    Start with the geometric series

    1/(1 - t) = Σ(n = 0 to ∞) t^n, convergent for |t| < 1.

    Differentiate both sides:

    1/(1 - t)^2 = Σ(n = 1 to ∞) nt^(n-1), still convergent for |t| < 1.

    Let t = -3x^2:

    1/(1 - (-3x^2)^2 = Σ(n = 1 to ∞) n(-3x^2)^(n-1), convergent for |-3x^2| = 3|x|^2 < 1.

    ==> 1/(1 + 3x^2)^2 = Σ(n = 1 to ∞) n(-3)^(n-1)x^(2n-2), convergent for |x| < 1/√3.

    Multiply both sides by x^3:

    x^3/(1 + 3x^2)^2 = Σ(n = 1 to ∞) n(-3)^(n-1)x^(2n+1), convergent for |x| < 1/√3.

    (radius of convergence is 1/√3)

    I hope this helps!

  • nle
    Lv 7
    8 years ago

    let f1 = 1 / (1 +u)^2

    find series of f1 then substitute u = 3x^2

    series converges when 3x^2 <1 or

    - sqrt(3)/3 < x < sqrt(3) /3

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