Question

Write the Taylor Series for f(x) = In (x)centered at a = 2. Show the first five terms.

Answer 1

Answer:

$f(x) = ln(2) + \frac{1}{2} (x-2) - \frac{1}{8} (x-2)^2 + \frac{1}{24} (x-2)^3 - \frac{1}{64} (x-2)^4 + ...$

Step-by-step explanation:

The Taylor Series for a function $f(x)$ centered at $x=a$ is:

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^{n}$

Let's find the first four derivatives of the function:

$f'(x) = \frac{1}{x}$

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

$f'''(x) = \frac{2}{x^3}$

$f^{iv}(x) = -\frac{6}{x^4}$

By replacing $x=2$ in the previous derivatives and then completing the expansion of the Taylor series of the function, we get:

$f(x) = f(2) + \frac{f'(2)}{1!} (x-2) + \frac{f''(2)}{2!} (x-2)^2 + \frac{f'''(2)}{3!} (x-2)^3 + \frac{f^{iv}(2)}{4!} (x-2)^4 + ...$

$f(x) = ln(2) + \frac{1}{2} (x-2) - \frac{1}{8} (x-2)^2 + \frac{1}{24} (x-2)^3 - \frac{1}{64} (x-2)^4 + ...$