Calculus Assignment 14

Taylor Series

For each function, only consider its valid ranges as indicated in the notes when you are computing the Taylor Series expansion.

The following is the formula for a Taylor Series expansion.

\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 +...+ \frac{f^n(a)}{n!}(x-a)^n\]

Question 1

\[ f(x) = \frac{1}{1-x}\]

Derivatives

We first evaluate the first few derivatives of f(x).

\[ f'(x) = \frac{1}{(1-x)^2}\] \[ f''(x) = \frac{2}{(1-x)^3}\] \[ f'''(x) = \frac{6}{(1-x)^4}\] \[ f^n(x) = \frac{n!}{(1-x)^{n+1}}\]

Use a = 0

Plugging in a = 0 into f(x), we get the following equation for the nth derivative:

\[ f^n(0) = n!\]

Using the standard formula for a taylor series expansion and the nth derivative formula for this function, we get the following series.

\[f(x) = 1 + x + x^2 +...+ x^{n}\]

Question 2

\[ f(x) = e^x\]

Derivatives

We first evaluate the first few derivatives of f(x).

\[ f'(x) = e^x\] \[ f''(x) = e^x\]

\[ f'''(x) = e^x\] \[ f^n(x) = e^x\]

Use a = 0

Plugging in a = 0 into f(x), we get the following equation for the nth derivative:

\[ f^n(0) = 1\]

Using the standard formula for a taylor series expansion and the nth derivative formula for this function, we get the following series.

\[ f^n(x) = 1 + x + \frac{x^2}{2!}+...+ \frac{x^n}{n!}\]

Question 3

\[ f(x) = ln(1+x)\]

Derivatives

We first evaluate the first few derivatives of f(x).

\[ f'(x) = \frac{1}{x+1}\] \[ f''(x) = \frac{-1}{(x+1)^2}\] \[ f'''(x) = \frac{2}{(x+1)^3}\]

\[ f^n(x) = \frac{(-1)^{n+1}(n-1)!}{(x+1)^n}\]

Use a = 0

Plugging in a = 0 into f(x), we get the following equation for the nth derivative:

\[ f^n(0) = (-1)^{n+1}(n-1)!\]

Using the standard formula for a taylor series expansion and the nth derivative formula for this function, we get the following series.

\[f(x) = x+ \frac{x^2}{2} +...+ \frac{(-1)^{n+1}}{n}x^n\]