Let f(x)=(1)/(1+x^2). (a) Find the Taylor series of f at a=0. (b) Prove that the Taylor series c

Let f(x)=(1)/(1+x^2). (a) Find the Taylor series of f at...

Key Concepts

Taylor Series

A Taylor series is an expansion of a function into an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. It provides a way to approximate functions locally by polynomials, enabling analysis and application in contexts where exact expressions are not easily manipulable.

Power Series

A power series is an infinite series expressed in terms of powers of a variable. It is a fundamental concept in mathematics and analysis, particularly used to represent functions within a radius of convergence. Power series are essential for studying local behavior and providing solutions to differential equations.

Radius of Convergence

The radius of convergence is a key concept that determines the interval in which a power series converges to the function it represents. It is determined by applying tests such as the ratio test or root test, and it delineates the boundary between the values of the variable for which the series converges and diverges.

Analytic Functions

An analytic function is one that can be locally represented by a convergent power series. The concept of analyticity guarantees not only infinite differentiability but also that the function agrees with its Taylor series expansion in a neighborhood around the point of expansion. This property is fundamental in complex analysis and various applications in mathematics.

Transcript

00:01 So they want to find the taylor series for f of x is equal to the absolute value of x when we center it at 1.

00:06 And they're going to want us to show that we have an infinite radius of convergence, but the taylor series won't converge for all x.

00:17 All right, so first let me go ahead and move this down here.

00:21 So something that will help us is we need to recall that the absolute value of x is truly a piecewise function.

00:26 So it's x when x is strictly greater than 0 or greater than or equal to 0, and then it's negative x when x is less than 0.

00:40 And if we were to take the derivative of this, remember the only point that it may differ at is going to be where these two functions meet up so x and negative x.

00:51 But everywhere else, so we just take the derivative of x, so that means it would be a derivative of 1 when x is strictly greater than 0.

00:58 And negative 1 when x is strictly less than 0.

01:02 And we know from just a calculus 1 class that the derivative of the absolute value at 0 doesn't exist.

01:11 So we don't need to worry about this because it's more of a sharp edge as opposed to a smooth piece.

01:18 All right.

01:19 So we have that.

01:21 And actually, if we were to just go ahead and take the derivative of this now, second derivative is just going to be 0 when x is strictly greater than 0 when x is strictly less than 0 and really if we think about it for any of the derivatives past this so i'll just say like n plus 2 for some in this is just going to be equal to 0 if x is not equal to 0 and then we'll just say undefined when x is equal to 0 all right so we'll just go ahead and use this derivatives here to actually fill out our tailor polynomial over here.

02:07 So let's see.

02:08 C is 1...

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