Maclaurin Series Representation of ln(cosx) and Its Derivative

In mathematical analysis, the Maclaurin series is a powerful tool for representing functions as series. This article will delve into the Maclaurin series representation of ln(cosx) and its derivatives.

The Maclaurin Series for ln(cosx)

Let's start by finding the Maclaurin series representation of ln(cosx). The Maclaurin series for a function can be found by expanding the function into a Taylor series around x0.

Step 1: Maclaurin Series for cosx

The Maclaurin series for cosx is given by:

cosx sum_{n0}^{infty} frac{-1^n x^{2n}}{2n!} 1 - frac{x^2}{2!} - frac{x^4}{4!} - frac{x^6}{6!} cdots

Step 2: Series Expansion for ln(1 - u)

The series expansion for ln(1 - u) is:

ln(1 - u) -sum_{n1}^{infty} frac{u^n}{n} quad text{for } |u|

Step 3: Substitute u 1 - cosx

Let u 1 - cosx. We know that:

cosx 1 - frac{x^2}{2} - frac{x^4}{24} - cdots

So, 1 - cosx frac{x^2}{2} - frac{x^4}{24} - cdots

We can express u as:

u frac{x^2}{2} - frac{x^4}{24} - Ox^6

Step 4: Substitute into ln(1 - u)

We want to find ln(cosx):

ln(cosx) ln(1 - (1 - cosx)) ln(1 - u)

Using the series expansion:

ln(cosx) -sum_{n1}^{infty} frac{1 - cosx^n}{n}

Step 5: Calculate the Series

Calculating 1 - cosx^n will involve binomial expansions and will lead to complicated higher-order terms. Instead, we can directly compute the first few terms of ln(cosx) using the known series expansion for small x:

ln(cosx) -frac{x^2}{2} - frac{x^4}{12} - frac{x^6}{45} - cdots

Step 6: Final Result

Thus, the Maclaurin series representation of ln(cosx) is:

ln(cosx) -sum_{n1}^{infty} frac{-1^{n-1} x^{2n}}{2n} text{ for small } x

This series converges for x .

The Derivative of ln(cosx)

The derivative of ln(cosx) is -tanx. The Maclaurin series for tanx is well-known and involves the so-called Bernoulli numbers. The series for tanx can be found as:

tanx x x^3/3 2x^5/15 17x^7/315 cdots

The coefficients involve Bernoulli numbers, and the series can be derived by integrating term by term and adjusting the constant of integration.

Conclusion

This article has explored the Maclaurin series representation of ln(cosx) and its derivative -tanx. Understanding and applying these series is crucial in advanced calculus and mathematical analysis. The series representation allows for the approximation of these functions with high accuracy, especially for small values of x.

Keywords: Maclaurin series, ln(cosx), tanx, Bernoulli numbers