Understanding the Maclaurin Series for ln(1 x)/(1-x)

The Maclaurin Series is a powerful tool in mathematics that allows us to express a function as an infinite sum of powers of a variable. It is particularly useful in approximation and simplification of complex functions. In this article, we explore the Maclaurin Series for the function ln(1 x)/(1-x).

rewrite and Simplification of the Function

To begin, we simplify the given function:

[ f(x) lnleft(frac{1 x}{1-x}right) ]

By applying the properties of logarithms, we can rewrite this function as:

[ f(x) ln(1 x) - ln(1-x) ]

Maclaurin Series for ln(1 x)

The Maclaurin series for ln(1 x) is given by:

[ ln(1 x) x - frac{x^2}{2} frac{x^3}{3} - frac{x^4}{4} cdots ]

This can be written in a more general form as:

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

Maclaurin Series for ln(1-x)

Similarly, the Maclaurin series for ln(1-x) is:

[ ln(1-x) -x - frac{x^2}{2} - frac{x^3}{3} - frac{x^4}{4} - cdots ]

This can be written in a more general form as:

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

Combining the Series

Now, we combine the two series to find the Maclaurin series for (lnleft(frac{1 x}{1-x}right)):

[ f(x) ln(1 x) - ln(1-x) ]

Substituting the series for ln(1 x) and ln(1-x):

[ f(x) left( x - frac{x^2}{2} frac{x^3}{3} - frac{x^4}{4} cdots right) - left( -x - frac{x^2}{2} - frac{x^3}{3} - frac{x^4}{4} - cdots right) ]

Combining like terms:

[ f(x) x - frac{x^2}{2} frac{x^3}{3} - frac{x^4}{4} x frac{x^2}{2} frac{x^3}{3} frac{x^4}{4} cdots ]

The negative and positive terms cancel each other out, leaving:

[ f(x) 2x 0 frac{2x^3}{3} 0 frac{2x^5}{5} cdots ]

This can be expressed in a more compact form as:

[ f(x) 2 sum_{n0}^{infty} frac{x^{2n 1}}{2n 1} quad text{for} quad |x|

Final Maclaurin Series for ln(1 x)/(1-x)

Thus, the Maclaurin series for the function (lnleft(frac{1 x}{1-x}right)) is:

[ lnleft(frac{1 x}{1-x}right) 2 sum_{n0}^{infty} frac{x^{2n 1}}{2n 1} quad text{for} quad |x|

Conclusion

The Maclaurin series is a fundamental concept in calculus and is widely used in various fields such as physics, engineering, and economics. Understanding how to derive and manipulate these series is essential for anyone working with complex functions in mathematical analysis.