Question

Find the first few terms of the Maclaurin series for each of the following functions and check your results by computer. $$\frac{e^{x}}{1-x}$$

Short answer

The Maclaurin series for \( \frac{e^x}{1-x} \) is \[ 1 + 2x + \frac{5}{2}x^2 + \frac{13}{6}x^3 + \cdots \]

Step-by-step solution

Understand the Maclaurin series

The Maclaurin series of a function is the Taylor series centered at 0. For a function f(x), it is given by:\[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \]

Express the Function as a Product

Given function is \(\frac{e^{x}}{1-x}\). Notice it can be seen as a product of two functions: \( e^x \) and \( \frac{1}{1-x} \).

Find the Maclaurin Series for each Part

The Maclaurin series expansion of \(e^x\) is:\[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \] The Maclaurin series expansion of \(\frac{1}{1-x}\) (a geometric series) is: \[ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots \]

Combine the Series

Multiply the series of \( e^x \) and \( \frac{1}{1-x} \) term by term up to the desired number of terms. For simplicity, let's find the first few terms. Using the product of series rule, we get: \[ \left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \right) \cdot \left(1 + x + x^2 + x^3 + \cdots \right) \]

Write the combined series terms

Multiply and collect the terms up to \( x^3 \):\[ = 1 + (x + x) + \left( \frac{x^2}{2!} + x^2 + x^2 \right) + \left( \frac{x^3}{3!} + x \cdot x^2 + x^3 \right) \] \[ = 1 + 2x + \left( \frac{1}{2} + 1 + 1 \right) x^2 + \left( \frac{1}{6} + 1 + 1 \right) x^3 \]

Simplify the Expression

Combine and simplify the coefficients:\[ = 1 + 2x + \frac{5}{2} x^2 + \frac{13}{6} x^3 + \cdots \]

Verify using a Computer

Use a computer algebra system (CAS) like WolframAlpha or a tool like Mathematica to verify the result by inputting the Maclaurin series expansion command.

Key concepts

Taylor series

The Taylor series is a powerful mathematical tool used to represent functions as infinite sums of their derivatives at a single point. Typically, the Taylor series for a function \( f(x) \) is given by:

\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \text{...} \]

In the equation above, \(a\) is the point around which the series is centered. When \(a = 0\), we call this specific series a Maclaurin series. It's essentially a special case of the Taylor series. Maclaurin series are especially useful for simplifying complex functions by representing them as polynomials. This series can be utilized to approximate functions near the point \(x = 0\).
The major advantage of using Taylor or Maclaurin series is the ease of dealing with complex functions, as polynomials are simpler to differentiate and integrate.

Geometric series

A geometric series is a series with a constant ratio between successive terms. It can be written as:

\[ a + ar + ar^2 + ar^3 + \text{...} \]

where \( a \) is the first term and \( r \) is the common ratio. One crucial geometric series that often appears in calculus and Maclaurin series expansions is the sum of \( \frac{1}{1-x} \), which can be written as:
\[ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \text{...} \]

This series is valid for \( |x| < 1 \). The geometric series is significant because it frequently appears in the process of finding Maclaurin series for various functions, as seen in the given exercise.

Series expansion

Series expansion involves expressing a function as a sum of simpler terms, typically polynomials. In the context of the Maclaurin series, each term's coefficient is derived from the function's derivatives at a single point (usually at 0). For example, given the function:

\( \frac{e^x}{1-x} \)

We can break it down into simpler components: \( e^x \) and \( \frac{1}{1-x} \). The Maclaurin series expansions for these are:
\[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \text{...} \] and
\[ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \text{...} \]

By multiplying these two series together and aligning powers of \( x \), we obtain the Maclaurin series for the original function.

Computer algebra system

A computer algebra system (CAS) is software designed to perform symbolic mathematical computations. These systems can handle algebraic expressions, calculus operations, matrix manipulations, and even series expansions. Examples of CAS include WolframAlpha, Mathematica, and Maple.

For the function \( \frac{e^x}{1-x} \), a CAS can be used to verify the manual calculation of its Maclaurin series. By entering the function into the CAS, the software can generate the series expansion up to any desired number of terms. This verification step is crucial, especially for complicated functions, to ensure the accuracy of the manual computations.
Utilizing a CAS can significantly save time and help verify results, reducing the risk of human error in complex symbolic operations.