Power Series Expansion for ( f(x) frac{1}{3-x} ) Centered at ( c 2 )

In this article, we will explore the power series expansion for the function ( f(x) frac{1}{3-x} ) centered at ( c 2 ). This process involves transforming the function into a form that can be expressed as a power series, which is a fundamental concept in both mathematics and computer science. We will prove that the series converges under certain conditions and derive the explicit form of the series.

Identifying the Power Series Representation

Given the function ( f(x) frac{1}{3-x} ), we want to find a power series representation centered at ( c 2 ). The general form of a power series centered at ( c ) is:

[ f(x) sum_{k0}^{infty} a_k (x-c)^k ]

First, let's manipulate the function ( f(x) frac{1}{3-x} ) to fit the form that allows us to express it as a power series.

Transformation to a Geometric Series

Consider the function rewritten as: [frac{1}{3-x} frac{1}{1 - (x-2 2-x)} frac{1}{1 - (x-2-1)}]

Here, we have separated the terms in such a way that it resembles the form of a geometric series:

[frac{1}{1 - (x-2-1)} frac{1}{2 - (x-2)}]

Now, let's denote ( x - c ) as a new variable ( y ), where ( c 2 ). Thus, we have:

[ y x - 2 ]

Substituting ( y ) into the function:

[ frac{1}{3 - (y 2)} frac{1}{1 - y} ]

The function ( frac{1}{1 - y} ) is a geometric series, which converges if and only if ( |y|

[ frac{1}{1 - y} sum_{k0}^{infty} y^k ]

Substituting back ( y x - 2 ), we get:

[ frac{1}{3 - x} frac{1}{1 - (x-2)} frac{1}{1 - (x-2)} sum_{k0}^{infty} (x-2)^k ]

Final Power Series Representation

The final power series representation for ( f(x) frac{1}{3-x} ) centered at ( c 2 ) is:

[ f(x) sum_{k0}^{infty} (x-2)^k ]

Now, let's verify the conditions for the convergence of the series. The series converges if and only if ( |x-2|

This means that the interval of convergence for the series is ( 1

Conclusion

In this article, we have successfully derived the power series representation for the function ( f(x) frac{1}{3-x} ) centered at ( c 2 ). The series is given by:

[ frac{1}{3-x} sum_{k0}^{infty} (x-2)^k ]

It is important to note that this series converges only within the interval ( (1, 3) ). Understanding this is crucial for both theoretical and practical applications in mathematics and computer science.

The process of finding a power series representation for a function is a powerful tool in analyzing and solving complex mathematical problems. By expanding functions into power series, we can gain deeper insights into their behavior and properties.