Understanding the Taylor Polynomial of x/(x-1) Around x2: A Comprehensive Guide
The Taylor polynomial is a powerful tool for approximating complex functions,especially when dealing with small perturbations around a specific point. In this article, we will delve into the Taylor polynomial of the function x/(x-1) around the point x2. We will explore how to calculate it and how to leverage the McLaurin series for this purpose.
Introduction to Taylor Polynomial and McLaurin Series
A Taylor polynomial of a function f(x) around a point a is a polynomial representation of the function in the neighborhood of that point. The general form of the first few terms of this polynomial is given by:
(P_n(x) f(a) f'(a)(x-a) frac{f''(a)}{2!}(x-a)^2 ... frac{f^{(n)}(a)}{n!}(x-a)^n)
When the point of expansion is x0, the resulting polynomial is known as the McLaurin series, simplifying the formula as:
(P_n(x) f(0) f'(0)x frac{f''(0)}{2!}x^2 ... frac{f^{(n)}(0)}{n!}x^n)
Problem Statement and Approach
We are tasked with finding the Taylor polynomial of the function x/(x-1) around the point x2. The goal is to leverage the McLaurin series as a stepping stone to solve this problem. Let's proceed step-by-step.
Step 1: Replace x with (x-2)
The first thing we note is that the given function can be rewritten in a form that makes it easier to work with the McLaurin series:
(frac{x}{x-1} frac{(x-2) 2}{(x-2)-1} -2 cdot frac{1}{1 - frac{x-2}{2}} -2 cdot sum_{n0}^{infty} left(frac{x-2}{2}right)^n) when (|frac{x-2}{2}|
Step 2: Transform the Function
The above expression is derived from the geometric series formula: (frac{1}{1-r} sum_{n0}^{infty} r^n), valid for |r| . Here, r frac{x-2}{2}, which gives us:
(frac{x}{x-1} -2 cdot sum_{n0}^{infty} left(frac{x-2}{2}right)^n)
Step 3: Derive the Taylor Polynomial
The McLaurin series for x/(x-1) around x2 is now directly given by replacing x with (x-2). We can see that the terms of the series essentially provide the Taylor polynomial, where each coefficient is scaled by -2 and the powers of (x-2) are as follows:
(frac{x}{x-1} -2 [1 frac{1}{2}(x-2) frac{1}{2^2}(x-2)^2 frac{1}{2^3}(x-2)^3 ...])
This means that the Taylor polynomial for the function around x2 can be written as:
(P_n(x) -2 sum_{n1}^{infty} (-1)^n cdot frac{(x-2)^n}{2^{n-1}})
Conclusion
In this article, we have dissected the problem of finding the Taylor polynomial of the function x/(x-1) around x2. We leveraged the McLaurin series and the method of replacing the variable to simplify the calculation. The key takeaway is the formulation of the polynomial series and understanding the application of the Taylor series in such scenarios.
If you are still struggling with this concept, feel free to ask for more detailed walkthroughs and tutorials. Happy learning!