Taylor series and interval of convergence


c. Determine the interval of convergence of the series.


f(x) = 1/x², a=1

Verified step by step guidance

1. Start by writing the Taylor series expansion of the function \(f(x) = \frac{1}{x^2}\) centered at \(a = 1\). This involves finding the derivatives of \(f(x)\) evaluated at \(x=1\) and expressing the series as \(\sum_{n=0}^{\infty} \frac{f^{(n)}(1)}{n!} (x-1)^n\).
2. Calculate the first few derivatives of \(f(x) = x^{-2}\) to identify a general pattern for \(f^{(n)}(x)\). For example, \(f'(x) = -2x^{-3}\), \(f''(x) = 6x^{-4}\), and so on. Then evaluate these derivatives at \(x=1\).
3. Express the general term of the Taylor series using the pattern found for \(f^{(n)}(1)\) and write the series explicitly as \(\sum_{n=0}^{\infty} c_n (x-1)^n\) where \(c_n = \frac{f^{(n)}(1)}{n!}\).
4. To determine the interval of convergence, apply the Ratio Test to the general term \(c_n (x-1)^n\). Compute the limit \(L = \lim_{n \to \infty} \left| \frac{c_{n+1} (x-1)^{n+1}}{c_n (x-1)^n} \right|\) and simplify it to an expression involving \(|x-1|\).
5. Find the values of \(x\) for which the Ratio Test limit \(L < 1\). This inequality will give the radius of convergence and thus the interval of convergence centered at \(x=1\). Finally, check the endpoints of the interval separately to determine if the series converges there.

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 a. It approximates the function near that point, using powers of (x - a). For f(x) = 1/x² at a = 1, the series involves derivatives evaluated at x = 1.

Radius and Interval of Convergence

The radius of convergence is the distance from the center a within which the Taylor series converges to the function. The interval of convergence is the set of x-values for which the series converges. 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 where a power series converges. These tests analyze the limit of the ratio or nth root of successive terms to find the radius of convergence, which is essential for identifying the interval where the Taylor series accurately represents the function.