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From this conclude that summation (2n choose n) x^n converges to 1/sqrt(1-4x) for x in (-1/4,1/4).

Then show that summation (2n choose n) (-1/4)^n = 1/sqrt(1-4(-1/4)) = 1/sqrt(2)

What I know:

Taylor series about 0 of f(x) = (1+x)^r, r is a real number given by summation (r choose n) x^n.

I know that (r choose n) can be rewritten as r(r-1)(r-2)..(r-n+1)/n!

and I know from a previous question that the Taylor series converges to f(x) for all x in (-1,1), and summation (2n choose n) x^n converges conditionally at x=-1/4.

How can I do this question with all this information? I am not sure how to piece it all together I am having a lot of trouble with this course.