This week, we’ll work out some Taylor Series expansions of popular functions.
- 1. f(x) = \(\frac {1}{1-x}\)
Solution
\[f(x) = \frac {1}{1-x}\] Formula for a Taylor Series Expansion: \[f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} (x - a) ^ {n}\] \[f(x)= f(a) + f'(a)(x-a) + \frac {f''(a)}{2!}(x-a)^2 + \frac {f'''(a)}{3!}(x-a)^3 + \frac {f''''(a)}{4!}(x-a)^4 + ........\] Derivatives of first order, second order, third order and fourth order of f(x) shown below:
\[f(x) = \frac {1}{1-x}, f(0) = \frac {1} {1-0} = 1\] \[f'(x) = \frac {1}{(1-x)^2}, f'(0) = \frac {1} {(1-0)^2} = 1\] \[f''(x) = \frac {2}{(1-x)^3}, f''(0) = \frac {2} {(1-0)^3} = 2\] \[f'''(x) = \frac {6}{(1-x)^4}, f'''(0) = \frac {6} {(1-0)^4} = 6\] \[f''''(x) = \frac {24}{(1-x)^5}, f''''(0) = \frac {24} {(1-0)^5} = 24\] if we do till, \[f^n(x) = \frac {n!}{(1-x)^{(n+1)}}\]
Substitute expressions into Taylor Series expansion:
\[f(x) = 1 + \frac {1}{1!}x^1 + \frac {2}{2!}x^2 + \frac {6}{3!}x^3 + \frac {24}{4!}x^4 + .....+ \frac {n!}{n!}x^n\] \[f(x) = \sum_{n = 0}^{\infty}x^n\]
- 2. f(x) = e^x
Solution
\[f(x) = e^x\] \[f'(x) = e^x, f'(0) = e^0 = 1\] \[f''(x) = e^x , f''(0) = e^0 = 1\] \[f'''(x) = e^x , f'''(0) = e^0 = 1\] \[.....\]
\[f^{(n)}(x) = e^x , f^n(0) = e^0 = 1\] Substitute expressions into Taylor Series expansion:
\[f(x) = \sum_{n=0}^{\infty} \frac {f^{(n)}(a)}{n!} (x - a) ^ {n}\]
\[f(x) = e^0 + e^0(x-0) + e^0(x-0)^2 + e^0(x-0)^3 + ...... + e^0(x-0)^n\]
\[f(x) = 1 + x + \frac {x^2}{2!} + \frac {x^3}{3!} + \frac {x^4}{4!} + ......+ \frac {x^n}{n!}\]
\[f(x) = \sum_{n=0}^{\infty} \frac {1}{n!}x^n\]
\[f(x) = \sum_{n=0}^{\infty} \frac {x^n} {n!}\]
- 3. f(x) = ln(1 + x)
Solution
\[f(x) = ln(1 + x), f(0) = 0\] \[f'(x) = \frac {1}{1+x}, f'(0) = 1\]
\[f''(x) = \frac {-1}{(1+x)^2}, f''(0) = -1\] \[f'''(x) = \frac {2}{(1+x)^3}, f'''(0) = 2\]
\[f''''(x) = \frac {-6}{(1+x)^4}, f'''(0) = -6\] \[......\]
\[f^n(x) = \frac {(-1) ^{(n-1)}(n-1)!} {(x+1)^n}, f^n(0) = (-1) ^{(n-1)}(n-1)!\]
Substitute expressions into Taylor Series expansion:
\[f(x) = f(0) + f'(0)(x-0) + \frac {f''(0)}{2!}(x-0)^2 + \frac {f'''(0)}{3!}(x-0)^3 + \frac {f''''(0)}{4!}(x-0)^4 + ........\]
\[f(x) = 0 + x - \frac {1}{2}x^2 + \frac{1}{3}x^3 - \frac {1}{4}x^4+ ......\]
\[f(x) = \sum_{n=1}^{\infty} \frac {(-1) ^{(n-1)}(n-1)!} {n!} (x)^n\]