## Example 8.4.7 (c): Using Taylor's Theorem

If the function

*f(x) =*had a Taylor series centered at*c = 0*, what would be its radius of convergence?If the function had a Taylor series, the remainder would go to zero and
the function would be infinitely often differentiable. It is clear that
*f(x) = * is (infinitely often)
differentiable for *x > -1*. Therefore the Taylor series centered at
*c = 0* is not expected to converge at *x = -1*.
Therefore our guess for the radius of convergence is:

r = 1

For your enjoyment, the function does have a Taylor series and you can double-check that:

f(0) = 1,

f'(0) = 1/2,

for n > 1

Above you see how well the sixth-degree Taylor polynomial approximates the square-root function.