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. $$(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots, \text { for }-1 < x < 1$$ $$\frac{1}{\left(1+4 x^{2}\right)^{2}}$$

Short answer

Answer: The first four nonzero terms are $$1,\, -8x^2,\, 48x^4,\, -256x^6$$.

Step-by-step solution

Substitute and Simplify

Substitute $$4x^2$$ for $$x$$ using the Taylor series of $$f(x) = (1+x)^{-2}$$: $$\frac{1}{\left(1+4 x^{2}\right)^{2}} = (1 + 4x^2)^{-2} = 1 - 2(4x^2) + 3(4x^2)^2 - 4(4x^2)^3 + \cdots$$ Now, expand and simplify the terms: $$1 - 2(4x^2) + 3(16x^{4}) - 4(64x^{6}) + \cdots = 1 - 8x^2 + 48x^4 - 256x^6 + \cdots$$

Identify the First Four Nonzero Terms

Now that we have the Taylor series of the given function, we can identify the first four nonzero terms: $$1,\quad -8x^2, \quad48x^4, \quad-256x^6$$ So, the first four nonzero terms of the Taylor series centered at 0 for the given function are: $$\frac{1}{\left(1+4 x^{2}\right)^{2}} = 1 - 8x^2 + 48x^4 - 256x^6 + \cdots$$

Key concepts

Power Series

A power series is an important mathematical concept that can be thought of as an infinite polynomial. It takes the form \( \sum_{n=0}^{\infty} a_n (x-c)^n \), where \(a_n\) are coefficients, \(x\) is the variable, and \(c\) is the center of the series. When a power series is centered at \(c = 0\), it becomes a Taylor series centered at zero.Power series are used to approximate functions and calculate values that might be difficult to compute otherwise. They can represent both polynomials and certain kinds of non-polynomial functions. By evaluating several terms in a series, we can arrive at an approximation of a function's output for given inputs, especially useful for making predictions in calculus or analytical studies.

• Example: The geometric series \( \sum_{n=0}^{\infty} x^n = \frac{1}{1-x} \) when \(|x| < 1\).
• Functions like \(\sin x\), \(\cos x\), and \(e^x\) have power series representations.

Series Expansion

Series expansion is a method used in calculus to express a function as an infinite sum of terms. This process is used to simplify complex functions into a series of polynomial terms.A common type of series expansion is the Taylor series, which approximates a function around a specific point by using the derivatives of that function. These expansions allow complex functions to be expressed with simpler polynomial terms, making them easier to work with in many scenarios.For smaller values of \(x\), a finite number of terms can be enough to provide a fairly accurate approximation of the original function. That is why the question asks for the first four non-zero terms: these terms often provide a good estimation near the expansion point.

• Example: Expressing \(\ln(1+x)\) as \(x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\).
• Widely applied in physics, engineering, and economics to simplify models.

Calculus

Calculus forms the foundation for power series and series expansions. It involves two main branches: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which represents the rate of change of a function. Understanding derivatives is crucial to finding Taylor series, as each term in a Taylor series is based on the function's derivatives evaluated at the center of the series. Integral calculus, meanwhile, is concerned with the accumulation of quantities and is often paired with differential calculus to solve complex problems in functions and curves. Together, these tools allow us to manipulate and approximate functions effectively using series expansions.

• Taylor series expansions derive from calculus through repeated differentiation of functions.
• Many calculus problems require the application of series expansions for solutions.

Mathematical Substitution

Mathematical substitution is a technique where we replace a variable or expression within a function to make calculations more manageable.This is commonly used in solving series problems, such as in our given exercise where \(4x^2\) is substituted into the Taylor series \( (1+x)^{-2} \). Through substitution, you replace \(x\) with \(4x^2\) to change the series’ context to the new function \( (1+4x^2)^{-2} \).Substitution is invaluable in mathematics as it helps simplify expressions, integrate wealthier functions, and transform integrals into calculable series.

• Essential for solving differential equations and integrals.
• Facilitates the conversion of complex expressions into straightforward problems.