Find the Taylor series for the functions defined as follows. Give the interval of convergence fo

step 1 Hello, we are looking at chapter 12, section 5, problem number 21. And we are to try to find the

Find the Taylor series for the functions defined as follows....

Key Concepts

Taylor Series

A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the derivatives of the function at a single point. This method allows one to approximate a function near that point by a polynomial, and it is particularly useful in situations where the function itself may be too complex to analyze directly.

Series Expansion for ln(1+u)

The function ln(1+u) can be expanded into a Taylor series around u = 0 as ln(1+u) = u - u²/2 + u³/3 - u?/4 + ..., which converges for |u| < 1. This expansion is a standard result in calculus and serves as a building block for the Taylor series of more complex logarithmic functions.

Substitution in Series

When applying the Taylor series expansion to a function like ln(1+2x?), one performs an appropriate substitution by letting u = 2x?. This substitution converts the original function into the form required to use the known series for ln(1+u), allowing the direct application of the series formula and simplifying the process of finding the Taylor series.

Interval of Convergence

The interval of convergence of a Taylor series is the range of values for which the series converges to the function. After substituting u = 2x? in the series for ln(1+u), the condition |2x?| < 1 must be satisfied, which leads to an inequality that determines the values of x for which the series converges. This concept is essential for ensuring that the series representation is valid within the derived bounds.

Transcript

00:01 Hello, we are looking at chapter 12, section 5, problem number 21.

00:08 And we are to try to find the taylor series expansion for the given function, f of x equals natural log of 1 plus 2x to the 4th, and give the interval of convergence.

00:21 So i'm going to start with the elementary series, taylor series expansion that most resembles this, which would be our basic natural log expansion, which is the natural log of 1 plus x.

00:37 And that expansion starts with the first term x.

00:42 And then we have minus x squared over 2, and then plus x cubed over 3, minus.

00:54 So we have an alternating series.

00:58 And if we go to the nth term, we can write that as negative 1 to the n to account for the alternating, and then we have x to the n over n.

01:14 And this is true for the interval of negative 1 less than x, less than 1.

01:22 So we need to change this.

01:24 This is a pretty quick change.

01:27 We can simply replace this x with 2x to the 4th in one shot.

01:33 So we're going to write that we're going to replace every x with 2x to the 4th.

01:50 So that we'll wind up with the expansion for the natural log of 1 plus 2x to the 4th.

02:00 And so we're going to shoot through our elementary series, and everywhere we see an x, we're going to replace it with 2x to the 4th.

02:07 So starting with that original x for the first term.

02:11 So we'll say 2x to the 4th.

02:13 Oops, minus...

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