To find the Taylor series for the function $f(x) = \ln(x)$ centered at $a = 2$, we need to compute the derivatives of $f(x)$ at $x = 2$ and then express the series in the form:

$f(x) = \ln(2) + \sum_{n=1}^{\infty} \frac{f^{(n)}(2)}{n!} (x - 2)^n$

Step 1: Compute the derivatives of $f(x) = \ln(x)$

1. First derivative: $f'(x) = \frac{1}{x}$ At $x = 2$: $f'(2) = \frac{1}{2}$

2. Second derivative: $f''(x) = -\frac{1}{x^2}$ At $x = 2$: $f''(2) = -\frac{1}{4}$

3. Third derivative: $f^{(3)}(x) = \frac{2}{x^3}$ At $x = 2$: $f^{(3)}(2) = \frac{2}{8} = \frac{1}{4}$

4. Fourth derivative: $f^{(4)}(x) = -\frac{6}{x^4}$ At $x = 2$: $f^{(4)}(2) = -\frac{6}{16} = -\frac{3}{8}$

Step 2: Write the Taylor series

Now, substitute these derivatives into the Taylor series formula:

$f(x) = \ln(2) + \sum_{n=1}^{\infty} \frac{f^{(n)}(2)}{n!} (x - 2)^n$

Substituting the computed derivatives:

$f(x) = \ln(2) + \left(\frac{1}{2}(x - 2)\right) - \left(\frac{1}{4 \cdot 2}(x - 2)^2\right) + \left(\frac{1}{4 \cdot 3!}(x - 2)^3\right) - \left(\frac{3}{8 \cdot 4!}(x - 2)^4\right) + \cdots$

Simplifying, we get:

$f(x) = \ln(2) + \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x - 2)^n}{n \cdot 2^n}$

Step 3: Radius of Convergence $R$

The radius of convergence $R$ of the Taylor series can be found using the formula:

$\frac{1}{R} = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

Here, the series is:

$\sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x - 2)^n}{n \cdot 2^n}$

Thus, the radius of convergence $R$ is:

$R = 2$

Final Answers:

1. Taylor Series: $f(x) = \ln(2) + \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x - 2)^n}{n \cdot 2^n}$

2. Radius of Convergence: $R = 2$

Would you like any additional details or have any questions?

Here are some related questions you might find interesting:

1. How do you derive the formula for the radius of convergence using the ratio test?
2. What happens to the Taylor series if we expand around $a = 1$ instead?
3. Can you calculate the first few terms of the Taylor series expansion for a different function, like $e^x$?
4. How does the error term $R_n(x)$ behave in Taylor series approximations?
5. What are some practical applications of Taylor series in physics or engineering?
6. How can you use Taylor series to approximate $\ln(2.1)$?
7. How does the Taylor series relate to the concept of analytic functions?
8. What are the conditions under which a function has a Taylor series expansion?

Tip: The Taylor series is especially useful for approximating functions near a point where the function is difficult to evaluate directly.