How to Find Taylor Series Coefficients Using Known Series Expansion

When dealing with complex functions, determining the coefficients (a_k) in the Taylor series can be challenging. However, one efficient method involves breaking down the function into simpler components, leveraging known series expansions. This article illustrates the process of finding the 88th derivative of a rational function at (x 0) using the concept of partial fractions and known series.

Breaking Down the Function

The given function is:

[f(x) -frac{x^2 x 1}{x - 2x^2 - 3}]

We start by factoring the denominator:

[-x^3 - 2x^2 - 3x - 6 -x(2 x^2 3)]

Thus, we can rewrite the function as:

[f(x) -frac{x^2 x 1}{x - 2x^2 - 3} frac{x^2 x 1}{2 - x - 3x^2}]

Further, we use partial fractions:

[f(x) frac{1}{2 - x} - frac{1}{3x^2}]

Using Geometric Series to Find the Series Expansion

Now, we expand each rational function using the geometric series:

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

For the first term:

[frac{1}{2 - x} frac{1}{2} cdot frac{1}{1 - frac{x}{2}} frac{1}{2} sum_{n0}^{infty} left(frac{x}{2}right)^n frac{1}{2} sum_{n0}^{infty} frac{x^n}{2^n}]

For the second term:

[frac{1}{3x^2} -frac{1}{3} cdot frac{1}{1 - left(-frac{x^2}{3}right)} -frac{1}{3} sum_{n0}^{infty} left(-frac{x^2}{3}right)^n -frac{1}{3} sum_{n0}^{infty} frac{(-1)^n x^{2n}}{3^n}]

Combining these, we get:

[f(x) frac{1}{2} sum_{n0}^{infty} frac{x^n}{2^n} - frac{1}{3} sum_{n0}^{infty} frac{(-1)^n x^{2n}}{3^n}]

Thus, the Taylor series expansion of (f(x)) is:

[f(x) sum_{n0}^{infty} frac{1}{2^{n 1}} x^n - sum_{n0}^{infty} frac{(-1)^{n 1}}{3^{n 1}} x^{2n}]

Finding the 88th Derivative at (x 0)

In a Taylor series expansion, the coefficient (a_k) of (x^k) is given by:

[a_k frac{f^{(k)}(0)}{k!}]

To find the 88th derivative at (x 0), we set (n 88) in our series expansions. For the first series, we get:

[frac{f^{(88)}(0)}{88!} frac{1}{2^{89}}]

For the second series, we set (n 44):

[frac{f^{(88)}(0)}{88!} -frac{1}{3^{45}}]

Equating these, we solve for (f^{(88)}(0)):

[f^{(88)}(0) 88! cdot left(frac{1}{2^{89}} - frac{1}{3^{45}}right)]

[boxed{f^{88}(0) 88! cdot left(frac{1}{2^{89}} - frac{1}{3^{45}}right)}]