Using Maclaurin Series to Find Non-Zero Terms of ln cos(4x)

In this article, we will delve into the method of using the Maclaurin series expansion to find the first four non-zero terms of the function ln cos(4x). The Maclaurin series expansion is a powerful tool in mathematics that allows us to express complex functions as infinite series of simpler terms. Let's dive right in!

Understanding the Maclaurin Series

The Maclaurin series for a function f(x) is given by the following formula:

fx Σn0∞ (f(n)(0)/n!) xn Σn0∞ Mn

This expansion helps us approximate the function ln cos(4x) with a series of terms. We will focus on finding the first four non-zero terms of this series.

Step-by-Step Solution

Step 1: Start with the Function

We begin with the function ln cos(4x).

ln cos(4x) f(x)

Step 2: Compute the Derivatives at x 0

To find the first non-zero term, we start by taking the derivatives of ln cos(4x) and evaluating them at x 0.

First Derivative

fx -4 sin(4x)/cos(4x) -4 tan(4x)

When x 0, fx 0.

Second Derivative

fx -16 sec2(4x)

When x 0, fx -16.

Third Derivative

fx 128 sec2(4x) tan(4x)

When x 0, fx 0.

Fourth Derivative

fx 128 (2 tan3(4x) tan(4x)) sec4(4x)

When x 0, fx -128.

The first non-zero term comes from the second derivative.

Step 3: Construct the Maclaurin Series

The second derivative term gives us the first non-zero term:

M2 -8x2

We can now construct the first few terms of the Maclaurin series expansion for ln cos(4x) by continuing to take derivatives and evaluating them at x 0.

Step 4: Higher-Order Terms

By continuing the process, we can find the next non-zero terms. Here are the first four non-zero terms of the series:

ln cos(4x) -8x2 - 64x4/3 - 1024x6/45 - ...

Understanding Taylor Polynomials

To better understand the series, let's derive the Maclaurin series for cos(x) and ln(1 x).

cos(x) 1 - x2/2! x4/4! - ...

ln(1 x) x - x2/2 x3/3 - ...

Combining these series and ignoring higher-degree terms can help us simplify the expression for ln cos(4x).

Alternative Approach Using Derivatives of tan(x)

Another approach is to use the fact that ln cos(x) is the antiderivative of -tan(x). We can find the Maclaurin series for tan(x) and then integrate it to find the Maclaurin series for ln cos(x).

tan(x) x x3/3 2x5/15 17x7/315 ...

By multiplying both sides by -1 and integrating from 0 to x, we get:

ln cos(x) -x2/2 - x4/12 - x6/45 - 17x8/2520 ...

Substitute 4x into this series to find the Maclaurin series for ln cos(4x).

These methods provide a systematic way to find the first few non-zero terms of the Maclaurin series expansion of ln cos(4x).