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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
- kbLv 78 years agoFavorite 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!
- nleLv 78 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