Interactive Real Analysis

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 8.1. Pointwise Convergence
• 8.2. Uniform Convergence
• 8.3. Series and Power Series
• 8.4. Taylor Series
• 8.5. Approximation Theory
• 9. Historical Tidbits
• Java Tools

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?

Back

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.