Understanding the Taylor Series Expansion for ( frac{x^2}{1-x^3} )
To find the Taylor series expansion for the function ( frac{x^2}{1-x^3} ), we can use a combination of known series expansions and algebraic manipulation. This method is more efficient and direct than solving a recurrence relation.
Step-by-Step Breakdown of the Process
First, let's recall the geometric series formula for ( frac{1}{1-x} ) where (|x| [ frac{1}{1-x} sum_{n0}^{infty} x^n ]
By differentiating both sides, we can obtain the series for ( frac{1}{1-x^2} ).
1. **Differentiating the Geometric Series**
[ frac{d}{dx} left( frac{1}{1-x} right) frac{d}{dx} left( sum_{n0}^{infty} x^n right) ] [ frac{1}{(1-x)^2} sum_{n1}^{infty} n x^{n-1} sum_{n0}^{infty} (n 1) x^n ]2. **Differentiating Again**
[ frac{d}{dx} left( frac{1}{1-x^2} right) frac{2}{(1-x^2)^2} sum_{n1}^{infty} n(n 1) x^{n-2} sum_{n0}^{infty} (n 1)n x^{n-2} ] [ frac{1}{1-x^2} sum_{n0}^{infty} frac{(n 1)n}{2} x^n ]3. **Using Partial Fraction Decomposition**
[ frac{1x^2}{1-x^3} frac{1}{1-x} - 4 frac{1}{1-x^2} 4 frac{1}{1-x^3} ]This can be verified by expanding and combining the fractions.
Direct Approach: Expanding Known Series
Another approach is to use known series expansions and algebraic manipulation without partial fractions.
1. **Starting with the Geometric Series**
[ frac{1}{1-x} sum_{n0}^{infty} x^n ]2. **Differentiating Term-by-Term**
[ frac{1}{1-x^2} sum_{n1}^{infty} n x^{n-1} sum_{n0}^{infty} (n 1) x^n ] [ frac{2}{1-x^3} sum_{n2}^{infty} n(n-1) x^{n-2} sum_{n0}^{infty} frac{n(n-1)}{2} x^n ]3. **Multiplying by ( x^2 )**
We multiply the series expression for ( frac{1}{1-x^3} ) by ( x^2 ) to obtain:
[ frac{1x^2}{1-x^3} frac{1}{1-x} - 4 frac{1}{1-x^2} 4 frac{1}{1-x^3} ] [ sum_{n0}^{infty} x^n - 4 sum_{n0}^{infty} n x^n 4 sum_{n0}^{infty} frac{n(n-1)}{2} x^n ] [ sum_{n0}^{infty} left( 1 - 4n 2n(n-1) right) x^n ] [ sum_{n0}^{infty} 2n^2 x^n ]4. **Simplifying the Expression**
[ sum_{n0}^{infty} 2n^2 x^n ]This is the Taylor series expansion of ( frac{1x^2}{1-x^3} ) for ( |x|
Conclusion
Using known series expansions and algebraic manipulation is a more efficient way to find the Taylor series for ( frac{1x^2}{1-x^3} ). This method avoids the complexity of solving recurrence relations and directly leverages known series to construct the desired function's series expansion.