Taylor Series Expansion of 1/lnx about x2

Understanding the Taylor series expansion of mathematical functions is a fundamental skill in advanced calculus and analysis. The Taylor series of a function $f(x)$ about a point $a$ is given by:

$f(x) f(a) frac{f'(a)}{1!}(x-a) frac{f''(a)}{2!}(x-a)^2 frac{f'''(a)}{3!}(x-a)^3 cdots$

This series can be generalized as:

$f(x) sum_{n0}^{infty} frac{f^{n}(a)}{n!}(x-a)^n$

Let's explore the Taylor series expansion of the function $f(x) frac{1}{ln{x}}$ about $x 2$.

Step-by-Step Solution

To find the Taylor series expansion of $f(x) frac{1}{ln{x}}$ about $x 2$, we need to calculate the derivatives of $f(x)$ and evaluate them at $x 2$. Here's a detailed step-by-step approach:

Step 1: Calculate $f(2)$

When $x 2$: $f(2) frac{1}{ln{2}}$

Step 2: Calculate the First Derivative and Its Value at $x2$

Using the chain rule, we get the first derivative:

$f'(x) -frac{1}{x(ln{x})^2}$

Evaluating at $x 2$: $f'(2) -frac{1}{2(ln{2})^2}$

Step 3: Calculate the Second Derivative and Its Value at $x2$

To find the second derivative, we use the product rule:

$f''(x) frac{2}{x^2(ln{x})^2} - frac{2}{x^2(ln{x})^3}$

Evaluating at $x 2$: $f''(2) frac{2}{4(ln{2})^2} - frac{2}{4(ln{2})^3} frac{1}{2(ln{2})^2} - frac{1}{2(ln{2})^3}$

Step 4: Generalize the Taylor Series Expansion

Now we can assemble the Taylor series expansion around $x 2$ using the derivatives calculated:

$f(x) frac{1}{ln{2}} - frac{1}{2(ln{2})^2} (x-2) frac{1}{2(ln{2})^2} - frac{1}{2(ln{2})^3} cdot frac{(x-2)^2}{2} cdots$

This series will continue with higher-order terms based on further derivatives evaluated at $x 2$.

Conclusion

The Taylor series expansion of $frac{1}{ln{x}}$ about $x 2$ is:

$f(x) frac{1}{ln{2}} - frac{1}{2(ln{2})^2} (x-2) frac{1}{2(ln{2})^2} - frac{1}{2(ln{2})^3} cdot frac{(x-2)^2}{2} cdots$

For a practical approach, one can use a Computer Algebra System (CAS) like Mathematica to compute the first few terms. Using Mathematica, the `FullSimplify` and `Series` functions can calculate the power series expansion:

FullSimplify[Series[1/Log[x], {x, 2, 4}]]

This command calculates the first five terms of the series expansion for $frac{1}{ln{x}}$ about $x2$.

This example demonstrates the application of Taylor series in analyzing complex functions and their behavior around specific points. Understanding these concepts is crucial for various applications in mathematics, physics, and engineering. Using tools like Mathematica can streamline these calculations and provide deeper insights into the function's properties.