Find the first few terms of the Maclaurin series for each of the following functions and check your results by computer. $$\frac{2 x}{e^{2 x}-1}$$

The Maclaurin series is a special type of Taylor series that simplifies the process of representing functions in an infinite sum of terms. It is centered at x = 0. The general formula for the Maclaurin series of a function f(x) is given by:
f(x) = f(0) + f'(0) x + \frac{f''(0) x^2}{2!} + \frac{f'''(0) x^3}{3!} + \frac{f^{(4)}(0) x^4}{4!} + ...
It is useful because it expresses complicated functions as an infinite polynomial, making them easier to work with. By breaking down the function into its derivatives evaluated at x = 0, the Maclaurin series captures the function's behavior close to this point.

In the context of our example, we are to find the Maclaurin series for the function \( \frac{2 x}{e^{2 x}-1} \). This involves calculating the derivatives, evaluating them at x = 0, and forming the polynomial.

To find the terms in the Maclaurin series, we must calculate the derivatives of our function.

Derivatives are crucial in capturing the rate of change of a function. Essentially, the nth derivative of a function provides us with the information about how the (n-1)th derivative changes over the function's input.
For instance, in our exercise, we start by calculating the first few derivatives of the function \( f(x) = \frac{2 x}{e^{2 x}-1} \). Here is the step-by-step process:

• First Derivative: f'(x)
• Second Derivative: f''(x)
• Third Derivative: f'''(x)

Each derivative will reveal more information about the function's behavior as we move further from x = 0.

After calculating the derivatives, the next step is to evaluate them at x = 0. This step helps us find the specific coefficients of the Maclaurin series.

To evaluate a derivative at a specific point means to substitute the point (in our case, x = 0) into the derivative. This simplifies the expression and provides a numerical value that can be used in the series.

For instance, we determine:

• f(0): evaluating the original function at x = 0
• f'(0): the first derivative at x = 0
• f''(0): the second derivative at x = 0

By doing so, each term contributes to the polynomial representation of the function around x = 0.

Symbolic calculations involve using algebraic symbols rather than numerical methods to solve problems. This is particularly useful for finding derivatives and evaluating series.

In our exercise, we might use symbolic calculation tools like Mathematica or a similar software. These can handle complex functions and provide accurate results quickly.

Using a computer as a verifying tool ensures that our manual calculations match the results provided by the software. For instance, we input \( \frac{2 x}{e^{2 x}-1} \) into the software and derive the Maclaurin series to cross-check our manual results.

This helps confirm the accuracy of our hand-calculated Maclaurin series.