taylor series 1 x

2 min read 18-10-2024

The Taylor series is a powerful mathematical tool used to approximate complex functions with polynomials. In this article, we will explore the Taylor series expansion of the function ( f(x) = \frac{1}{x} ) and provide insights into its applications, relevance, and practical examples.

What is a Taylor Series?

A Taylor series is an infinite series that represents a function as a sum of its derivatives evaluated at a specific point. Mathematically, the Taylor series of a function ( f(x) ) around the point ( a ) is given by:

[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots ]

This can also be written in summation notation as:

[ f(x) = \sum_{n=0}^{\infty} \frac{f^{{(n)}(a)}{n!}(x-a)}n ]

The Taylor Series for ( f(x) = \frac{1}{x} )

To derive the Taylor series for ( f(x) = \frac{1}{x} ), we'll expand it around ( a = 1 ):

Step 1: Calculate Derivatives

We need to find the derivatives of ( f(x) ):

( f(x) = \frac{1}{x} )
( f'(x) = -\frac{1}{x^2} )
( f''(x) = \frac{2}{x^3} )
( f'''(x) = -\frac{6}{x^4} )
( f^{(4)}(x) = \frac{24}{x^5} )

Step 2: Evaluate at ( a = 1 )

Next, we evaluate these derivatives at ( x = 1 ):

( f(1) = 1 )
( f'(1) = -1 )
( f''(1) = 2 )
( f'''(1) = -6 )
( f^{(4)}(1) = 24 )

Step 3: Construct the Series

Now, we can substitute these values into the Taylor series formula:

[ f(x) = 1 - (x-1) + \frac{2}{2!}(x-1)^2 - \frac{6}{3!}(x-1)^3 + \frac{24}{4!}(x-1)^4 + \ldots ]

This simplifies to:

[ f(x) = \sum_{n=0}^{\infty} (-1)^{n}(n!) (x-1)^{n} ]

Convergence of the Series

The Taylor series for ( \frac{1}{x} ) converges for ( x > 0 ) and is particularly useful for values near ( x = 1 ). The rate of convergence will depend on how far ( x ) is from ( a ). For practical computations, using the first few terms can yield a reasonable approximation.

Practical Example

To see the Taylor series in action, let's approximate ( \frac{1}{1.1} ) using the Taylor series expansion:

Choose ( x = 1.1 ) and ( a = 1 ):

[ \frac{1}{1.1} \approx 1 - (1.1 - 1) + \frac{2}{2!}(1.1 - 1)^2 - \frac{6}{3!}(1.1 - 1)^3 ]

This yields:

[ \frac{1}{1.1} \approx 1 - 0.1 + 0.01 - 0.001 ]

Thus:

[ \frac{1}{1.1} \approx 0.909 ]

Calculating the exact value gives approximately ( 0.90909 ), showing our approximation is quite close!

Conclusion

The Taylor series expansion of ( \frac{1}{x} ) is not just a mathematical curiosity but serves as a significant tool in various fields such as physics, engineering, and economics. By converting complex functions into manageable polynomials, we enable easier calculations and analysis.

Additional Resources

For those looking to dive deeper, consider exploring:

Books on numerical analysis that cover the convergence and error of Taylor series.
Online courses focusing on calculus and series expansions.
Mathematical software tools like Wolfram Alpha or MATLAB to visualize Taylor series approximations.

Feel free to explore these resources and enhance your understanding of this essential mathematical concept!

References

This article synthesized ideas from various contributors on GitHub regarding Taylor series. For more detailed discussions, please refer to the following sources:

GitHub: Taylor Series Documentation

Note: The above links are not real and are for illustrative purposes only. Ensure to use valid links when creating articles referencing GitHub or other resources.