Write the Taylor series for , centered at 0. Prove that the series converges to in (-1, 0]
By finding differentials of at x = 0, I wrote the Taylor series as
Sigma(n=0 to infinity) - (1/2^n) x^n
This is of form Sigma(n=0 to infinity) an x^n, where an = -(1//2^n)
Therefore it is a power series centered at 0
Let R = radius of convergence of the series.
1/R = lim(n->infinity) |an|^(1/n) = lim(n->infinity) (1/2^n)^(1/n) = 1/2
Or, R = 2
Therefore the series converges if x is in (-R, R) = (-2, 2)
(-1, 0] is a subset of (-2, 2)
Therefore the series converges if x is in (-1, 0]
So it is proved that it converges. But how to prove that the series Sigma(n=0 to infinity) -(1/2^n) x^n converges to sqrt(1-x) ?