Taylor series and interval of convergence


b. Write the power series using summation notation.


f(x) = ln (x − 2), a = 3

Verified step by step guidance

1. Identify the function and the center of the Taylor series expansion. Here, the function is \(f(x) = \ln(x - 2)\) and the center is \(a = 3\).
2. Rewrite the function in terms of \((x - a)\) to express it as a power series centered at \(x = 3\). Set \(u = x - 3\), so that \(x - 2 = (x - 3) + 1 = u + 1\).
3. Recall the Taylor series expansion for \(\ln(1 + u)\) around \(u = 0\), which is \(\ln(1 + u) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{u^n}{n}\) for \(|u| < 1\).
4. Substitute back \(u = x - 3\) into the series to write \(f(x)\) as a power series centered at \(x = 3\): \(f(x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x - 3)^n}{n}\).
5. Express the final answer in summation notation: \(f(x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x - 3)^n}{n}\), which represents the power series expansion of \(\ln(x - 2)\) about \(x = 3\).

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), allowing complex functions to be expressed as polynomials.

Power Series and Summation Notation

A power series is an infinite series of the form Σ c_n (x - a)^n, where c_n are coefficients and a is the center. Summation notation concisely expresses this infinite sum, making it easier to manipulate and analyze the series.

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges to the function. Determining this interval involves testing the radius of convergence and checking endpoints to ensure the series accurately represents the function within that range.