Taylor series and interval of convergence


c. Determine the interval of convergence of the series.


f(x)=2/(1−x)³, a=0

Verified step by step guidance

1. Recognize that the function given is \(f(x) = \frac{2}{(1 - x)^3}\), and we want to find the interval of convergence of its Taylor series centered at \(a = 0\).
2. Recall that the Taylor series for \(\frac{1}{(1 - x)^n}\), where \(n\) is a positive integer, can be expressed using the generalized binomial series or by differentiating the geometric series multiple times.
3. Since \(f(x) = 2 \cdot \frac{1}{(1 - x)^3}\), start from the geometric series \(\frac{1}{1 - x} = \sum_{k=0}^\infty x^k\) which converges for \(|x| < 1\), and differentiate it twice to get the series for \(\frac{1}{(1 - x)^3}\).
4. Use the fact that differentiation of a power series term-by-term preserves the radius of convergence, so the radius of convergence remains \$1$ after differentiations.
5. Conclude that the interval of convergence for the Taylor series of \(f(x)\) centered at \$0\( is \)(-1, 1)\(, but check the endpoints \)x = -1\( and \)x = 1$ separately by substituting into the series to determine if the series converges or diverges at these points.

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. For f(x) centered at a=0, the series uses derivatives evaluated at 0 to approximate f(x) near that point. Understanding how to find the general term is essential for expressing the function as a power series.

Radius and Interval of Convergence

The radius of convergence defines the distance from the center within which the Taylor series converges to the function. The interval of convergence includes all x-values where the series converges, possibly including endpoints. Determining this interval often involves applying convergence tests like the ratio or root test.

Convergence Tests for Power Series

Tests such as the ratio test or root test help determine whether a power series converges for specific values of x. These tests analyze the limit of the ratio or nth root of successive terms to find the radius of convergence. Applying these tests is crucial to identify the interval where the Taylor series accurately represents the function.