Problem 54 - Calculus Early Transcendentals

Question

Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series. $$(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots, \text { for }-1 < x < 1$$ $$\left(x^{2}-4 x+5\right)^{-2}$$

Step-by-Step Solution

Answer

Answer: The first four nonzero terms of the Taylor series centered at 0 for the function (x^2-4x+5)^{-2} are -4x^6+48x^5+3x^4+360x^3.

Step 1: Identify the Substitution Needed

First, notice that we need to transform the function \((1+x)^{-2}\) to the function \(\left(x^{2}-4 x+5\right)^{-2}\). We can do this using substitution by finding \(x\) such that: $$1+x=x^2-4x+5$$ Subtract 1 from both sides to find the substitution \(x\): $$x=x^2-4x+4$$ Now we can see that this is a quadratic equation in x. The substitution to be made is into the given Taylor series is \(x \rightarrow u\), where \(u = x^2 - 4x + 4\). The function becomes: $$(1+u)^{-2}$$

Step 2: Apply the Substitution to the Taylor Series

Now we have the substitution, we can apply it to the given Taylor series for the function \((1+x)^{-2}\) to get the Taylor series for our desired function: $$(1+u)^{-2} = 1-2u+3u^{2}-4u^{3}+\cdots$$ Now, substitute \(u = x^2 - 4x + 4\): $$\left(x^2-4 x+5\right)^{-2}=1-2(x^2 - 4x + 4) + 3(x^2 - 4x + 4)^{2} - 4(x^2 - 4x + 4)^{3} +\cdots$$

Step 3: Expand and Simplify the Taylor Series

Expand and simplify the first four nonzero terms of the Taylor series: $$\left(x^2-4 x+5\right)^{-2}=1-2(x^2-4x+4)+3(x^4-8x^3+16x^2-32x+16)-4(x^6-12x^5+48x^4-96x^3+96x^2+64x-64)+\cdots$$ Simplify each term to get: $$\left(x^2-4 x+5\right)^{-2}=1-2x^2+8x-8+3x^4-24x^3+48x^2-96x+48-4x^6+48x^5-192x^4+384x^3-384x^2+256x-256+\cdots$$

Step 4: Combine Like Terms and Write the First Four Nonzero Terms

Next, we'll combine like terms and write out the first four nonzero terms of the Taylor series: $$\left(x^2-4 x+5\right)^{-2}=-4x^6+48x^5+3x^4+360x^3-34x^2-520x+361+\cdots$$ So, the first four nonzero terms in the Taylor series centered at 0 for \(\left(x^2-4 x+5\right)^{-2}\) are $$\boxed{-4x^6+48x^5+3x^4+360x^3}$$

Key Concepts

Power Series

A power series can be thought of as an infinite series sum of the form \( a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots \).Each term of a power series has coefficients corresponding to powers of \(x\). Power series are powerful tools in calculus because they can represent functions as sums.

The Taylor series, a type of power series, is particularly useful.It expresses a function as an infinite sum of terms calculated from the values of its derivatives at a single point.

In this exercise, the expansion of the series \((1+x)^{-2}\) is given:

• \( 1 - 2x + 3x^2 - 4x^3 + \cdots \)

When dealing with transformations and substitutions for other functions,power series allows us to start from simple series and modify them to suit our needs.

Substitution

Substitution is a method used to transform one expression into another by replacing a part of the expression with a different variable or expression.In this context, substitution helps us express a complex function in terms of a simpler form that can be expanded using a Taylor series.

The exercise requires substituting \(x^2 - 4x + 5\) into the existing power series for \((1+x)^{-2}\).This is achieved by finding \(u\) such that:

• \(1+u = x^2 - 4x + 5\)

The exercise uses substitution where \(u = x^2 - 4x + 4\).This simplifies the function to \((1+u)^{-2}\), allowing us to use the original power series.

By substituting \(u\) back, the expansion becomes much easier and mirrors the structure of the simpler initial Taylor series.

Factoring of Constants

Factoring of constants is a technique employed during the expansion and simplification of series terms to clearly identify and combine like terms.When expressions become complex, as in polynomial expansions, managing constants efficiently helps to keep the equations manageable.

For example, in the series derived from substitution:

• Constants multiply with the terms of the polynomial expressions like \(1 - 2(x^2 - 4x + 4) + \cdots\)

Factoring out constants can simplify the math, ensuring that similar powers of \(x\) are easy to combine.This is seen when expanding as:

• \( -2x^2 + 8x \ Rightarrow\) Extract \(-2\) and \(8\) respectively for clarity.

Once expanded, Constants are often factored out again to collapse terms back to their simplest forms.This efficiency is essential to expressing the final Taylor series with the first few non-zero terms.