Introduction to Maclaurin Series for Trigonometric Functions and Logarithms

The Maclaurin series is a powerful tool in calculus to represent functions as power series. This article will explore the Maclaurin series for the functions ln(cos 4x) and cos x. Specifically, we will find the first four non-zero terms of the series for each function, using both direct differentiation methods and combining Taylor polynomials.

Maclaurin Series for cos x

We start with the Maclaurin series for cos x. Recall that the Maclaurin series of a function f(x) is given by:

[f(x) sum_{n0}^{infty} frac{f^{(n)}(0)}{n!} x^n]

Applying this to cos x, we have:

[f(0) cos 0 1] [f'(x) -sin x quad text{so} quad f'(0) 0] [f''(x) -cos x quad text{so} quad f''(0) -1] [f'''(x) sin x quad text{so} quad f'''(0) 0] [f^{(4)}(x) cos x quad text{so} quad f^{(4)}(0) 1] [f^{(5)}(x) -sin x quad text{so} quad f^{(5)}(0) 0] [f^{(6)}(x) -cos x quad text{so} quad f^{(6)}(0) -1] [vdots]

Therefore, the Maclaurin series for cos x truncated to the first four non-zero terms is:

[cos x 1 - frac{x^2}{2!} - frac{x^4}{4!} - frac{x^6}{6!} cdots]

Maclaurin Series for ln(cos x)

Next, we explore the Maclaurin series for ln(cos x). Since ln(1) 0, we can start by finding the derivative of ln(cos x):

[frac{d}{dx} ln(cos x) -frac{sin x}{cos x} -tan x]

Thus, we need to find the Maclaurin series for (-tan x). Recall that the Maclaurin series for (tan x) is an odd function and can be expanded as:

[tan x x - frac{x^3}{3!} frac{x^5}{5!} - frac{x^7}{7!} cdots]

Multiplying by (-1), we get:

[-tan x -x - frac{x^3}{3!} - frac{x^5}{5!} - frac{x^7}{7!} - cdots]

To find the series for (ln(cos x)), we integrate (-tan x) from 0 to x:

[ln(cos x) int_{0}^{x} -tan t , dt -int_{0}^{x} left(t - frac{t^3}{3!} - frac{t^5}{5!} - cdotsright) dt]

Integrating term by term, we have:

[ln(cos x) -left(frac{x^2}{2} - frac{x^4}{4 cdot 3!} - frac{x^6}{6 cdot 5!} - frac{x^8}{8 cdot 7!} - cdotsright)]

Truncating to the first four non-zero terms, we get:

[ln(cos x) -frac{x^2}{2} - frac{x^4}{12} - frac{x^6}{180} - frac{x^8}{10080}]

Maclaurin Series for ln(cos 4x)

Finally, we consider the Maclaurin series for ln(cos 4x). Using the chain rule and the Maclaurin series for ln(cos x), we can write:

[ln(cos 4x) ln(1 - 16 sin^2 x)]

Using the series for (sin x), and substituting into the series for (ln(1 - 16 sin^2 x)), we can find the first four non-zero terms. However, the direct differentiation approach is more straightforward:

[frac{d}{dx} ln(cos 4x) -frac{4sin 4x}{cos 4x} -4tan 4x]

Expanding (-4tan 4x) using the Maclaurin series for (tan 4x), and then integrating, we can find the series for (ln(cos 4x)).

Thus, the first four non-zero terms of the Maclaurin series for (ln(cos 4x)) are:

[ln(cos 4x) -frac{(4x)^2}{2!} - frac{(4x)^4}{12} - frac{(4x)^6}{180} - frac{(4x)^8}{10080}]

Conclusion

This article has covered the Maclaurin series for the functions cos x and ln(cos 4x). It explored both direct differentiation methods and combining Taylor polynomials to find the first four non-zero terms. These series are fundamental tools in mathematical analysis, particularly in calculus, and provide deep insights into the behavior of these functions near the origin.

Related Keywords

Maclaurin series Taylor polynomials logarithmic function trigonometric functions