Interval of Convergence for Taylor and Maclaurin Series

Introduction to Taylor and Maclaurin Series

Taylor and Maclaurin series are special types of power series that represent functions as infinite sums of terms involving derivatives at a point. A Taylor series centered at c has the form ∑ a_n(x – c)ⁿ, while a Maclaurin series is simply a Taylor series centered at c = 0. Understanding the interval of convergence for these series is crucial because it tells you for which x values the series actually equals the function.

Taylor vs. Maclaurin: How the Center Affects Convergence

The interval of convergence for a Taylor series depends heavily on the chosen center c. For an entire function like e^x or sin(x), the series converges for all real x regardless of center. However, for functions with singularities, the center determines the radius of convergence. For example, the Maclaurin series for 1/(1 – x) converges only for |x| < 1, but its Taylor series centered at c = 2 converges for |x – 2| < 1 (since the singularity at x=1 is 1 unit away). This shift is explained in detail on the how-to guide.

Common Functions and Their Intervals

The table below compares the intervals of convergence for Maclaurin series (c = 0) versus Taylor series about a different center for three important functions. Notice how the interval moves and sometimes changes length.

Step-by-Step Calculation for Taylor Series

Finding the interval of convergence for a Taylor series follows the same process as any power series: apply the ratio or root test to the general term a_n(x – c)ⁿ to find the radius R, then test the endpoints. The formula page explains the ratio test in detail. For example, consider the Taylor series for e^x about c = 2: the coefficients are e²/n!, so the ratio test gives R = ∞. For ln(1+x) about c = 1, the series converges for |x – 1| < 2, and checking endpoints shows convergence at x = 3 but not at x = –1.

Why the Center Matters in Calculus

In calculus, Taylor series are used to approximate functions near the center. The interval of convergence tells you how far you can go before the approximation fails. Maclaurin series are often taught first because they are simpler, but real-world problems may require expansion about arbitrary points. Our FAQ addresses common questions about endpoint behavior and series approximation.

Frequently Asked Questions

Does every Taylor series have the same radius as the Maclaurin series?

No. Only for entire functions (analytic everywhere) does the radius stay infinite. For functions with singularities, the radius equals the distance from the center to the nearest singularity.

How do I know whether to use a Taylor or Maclaurin series?

Use Maclaurin if the function is smooth near 0 and you need an approximation near 0. Use a general Taylor series when you need accuracy near a different point.