What is the Maclaurin's Series Expansion for (f(x) frac{1}{1-x}) up to the Sixth Term?

It is well known that the Maclaurin's series expansion of a function (f(x) frac{1}{1-x}) can be derived through its Binomial expansion. This expansion is particularly useful and can be used as a basis to extend to a higher degree if needed.

The function (f(x) frac{1}{1-x}) can be expressed as a geometric series:

[frac{1}{1-x} sum_{n0}^{infty} x^n]

Provided that (x , this series converges, making it an ideal representation for the given function. This geometric series representation is quite intuitive when we recognize the absolute value of the common ratio (x) is less than 1.

Deriving the Series

Let's consider the derivatives of (f(x) frac{1}{1-x}) to derive the series expansion:

First Derivative

The first derivative of (f(x)) is:

[f'(x) frac{d}{dx}left(frac{1}{1-x}right) frac{1}{(1-x)^2}]

Second Derivative

The second derivative of (f(x)) is:

[f''(x) frac{d^2}{dx^2}left(frac{1}{1-x}right) frac{2!}{(1-x)^3}]

Note the pattern here:

[f^{(n)}(x) frac{n!}{(1-x)^{n 1}}]

Maclaurin Series Expansion

Using these derivatives, the Maclaurin series for (f(x) frac{1}{1-x}) is given by:

[f(x) f(0) frac{f'(0)}{1!}x frac{f''(0)}{2!}x^2 frac{f'''(0)}{3!}x^3 cdots]

Substituting the values for the derivatives, we get:

[f(x) 1 frac{1!}{1!}x frac{2!}{2!}x^2 frac{3!}{3!}x^3 frac{4!}{4!}x^4 frac{5!}{5!}x^5]

Up to the Sixth Term

Continuing the pattern, the expansion up to the sixth term is:

[f(x) 1 x x^2 x^3 x^4 x^5 x^6]

This series provides an accurate approximation of (f(x) frac{1}{1-x}) for values of (x .

Conclusion

The Maclaurin series expansion of (f(x) frac{1}{1-x}) not only shows the power of series in approximating functions but also highlights the relationship between calculus and series expansions. Understanding this concept is crucial in fields such as physics, engineering, and advanced mathematics where series approximations are often used to simplify complex functions.

Key Points

Maclaurin Series Expansion: A method to express a function as an infinite sum of terms based on its derivatives at a point, typically zero. Geometric Series: A series of the form (sum_{n0}^{infty} ar^n), where (a) is the first term and (r) the common ratio. Derivatives: Calculating derivatives provides the necessary terms to build the series and ensures accuracy in the approximation.

References:

Taylor Series Geometric Series Taylor Series | Khan Academy