8.4. Taylor Series

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.

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