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The question is:

Use the binomial series to expand the function as a power series. State the radius of convergence.

x/root of(4+x^2)

I did it like this:

x/root of (4+x^2)

= x(1/2)[1+(x^2/4)]^(-1/2)

then i find the power series of [1+(x^2/4)]^(-1/2),and get:

sum of (from 0, infinity)[(-1)^n (3/2-n)(x^(2n))]/[n!4^n]

and then i multiply the power series by x(1/2)

and get:

sum of(from 0,infinity)[(-1)^n(3/2-n)(x^(2n+1))]/[n!2^(2n+1)]

which is different from the solution..Did i do anything wrong? I wonder if i can express the power series with fractions in it..

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Something is wrong with your solution; specifically, the term (3/2-n) is unclear. It seems to me that you have erred when you simplified the binomial coefficients.I suggest that you carefully do (1+x)^{-1/2} and then substitute x^{2}/4 in place of x.