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Find the Binomial Series

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2 Answers

 
g(x) = (1+5x2)-4
g(x) = ∑{k=0, ∞}C(α, k) (5x2)k, where α = -4, and
 
C(α, k) = α(α-1)(α-2)...(α-k+1)/k!
The binomial series is
(1+u)n = ∑k=1 (n k) uk   for |u|<1,
where (n k) is the generalized binomial coefficient, n(n-1)(n-2)...(n-k+1)/k!
 
Therefore,
(1+5x2)-4 = ∑k=1 (-4 k) 5kx2k, where (-4 k) = (-4)(-5)(-6)...(-3-k)/k!
 
In particular, (-4 1) = -4, (-4 2) = (-4)(-5)/2!=10, (-4 3) = (-4)(-5)(-6)/3! = -20, so that the first three terms are:
(1+5x2)-4 = -4 (5)x2 +10(25)x4 -20(125)x6 +- ... = -20x2 + 250x4 - 2500x6 +- ...