Question 1 - TAYLOR SERIES

This week, we’ll work out some Taylor Series expansions of popular functions.

\(f(x) = \frac{1}{(1−x)}\)
\(f(x) = e^x\)
\(f(x) = ln(1 + x)\)
\(f(x)=x^\frac{1}{2}\)

For each function, only consider its valid ranges as indicated in the notes when you are computing the Taylor Series expansion. Please submit your assignment as an R- Markdown document.

Taylor Series expansion formula: \(f(x)=\sum_{n=0}^∞\frac{f^n(a)}{n!}(x−a)^n\)
where n is derivative number, \(f^n(a) = f'(a), f''(a)\), …

Steps:
- Find the first few derivatives
- Solve them for f(0)
- Plug them into the formula
- Solve and find patterns

\(f(x) = \frac{1}{(1−x)}\)

\(f(a) = \frac{1}{(1−a)}\)
\(f'(a) = \frac{1}{(1−a)^2}\)
\(f''(a) = \frac{2}{(1−a)^3}\)
\(f'''(a) = \frac{6}{(1−a)^4}\)
\(f''''(a) = \frac{24}{(1−a)^5}\)
\(f'''''(a) = \frac{120}{(1−a)^6}\)
\(f''''''(a) = \frac{720}{(1−a)^7}\)

\(f(0) = 1\)
\(f'(0) = 1\)
\(f''(0) = 2\)
\(f'''(0) = 6\)
\(f''''(0) = 24\)
\(f'''''(0) = 120\)
\(f''''''(0) = 720\)

at a = 0:
\(f(x)= \frac{f(0)}{n!}(x−0)^n + \frac{f'(0)}{n!}(x−0)^n + \frac{f''(0)}{n!}(x−0)^n + \frac{f'''(0)}{n!}(x−0)^n + \frac{f''''(0)}{n!}(x−0)^n + \frac{f'''''(0)}{n!}(x−0)^n + \frac{f''''''(0)}{n!}(x−0)^n + ...\)
\(f(x)= \frac{1}{0!}(x)^0 + \frac{1}{1!}(x)^1 + \frac{2}{2!}(x)^2 + \frac{6}{3!}(x)^3 + \frac{24}{4!}(x)^4 + \frac{120}{5!}(x)^5 + \frac{720}{6!}(x)^6 + ...\)
\(f(x)= 1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + ...\)
The Taylor Series expansion of \(f(x) = \frac{1}{(1−x)} = 1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + ...\)

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

\(f(a) = e^a\)
\(f'(a) = e^a\)
\(f''(a) = e^a\)
\(f'''(a) = e^a\)
\(f''''(a) = e^a\)
\(f'''''(a) = e^a\)
\(f''''''(a) = e^a\)

\(f(0) = 1\)
\(f'(0) = 1\)
\(f''(0) = 1\)
\(f'''(0) = 1\)
\(f''''(0) = 1\)
\(f'''''(0) = 1\)
\(f''''''(0) = 1\)

at a = 0:
\(f(x)= \frac{f(0)}{n!}(x−0)^n + \frac{f'(0)}{n!}(x−0)^n + \frac{f''(0)}{n!}(x−0)^n + \frac{f'''(0)}{n!}(x−0)^n + \frac{f''''(0)}{n!}(x−0)^n + \frac{f'''''(0)}{n!}(x−0)^n + \frac{f''''''(0)}{n!}(x−0)^n + ...\)
\(f(x)= \frac{1}{0!}(x)^0 + \frac{1}{1!}(x)^1 + \frac{1}{2!}(x)^2 + \frac{1}{3!}(x)^3 + \frac{1}{4!}(x)^4 + \frac{1}{5!}(x)^5 + \frac{1}{6!}(x)^6 + ...\)
\(f(x)= \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!} + ...\)
\(f(x)= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!} + ...\)
The Taylor Series expansion of \(f(x) = e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!} + ...\)

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

\(f(a) = ln(1 + a)\)
\(f'(a) = \frac{1}{1 + a}\)
\(f''(a) = -\frac{1}{(1 + a)^2}\)
\(f'''(a) = \frac{2}{(1 + a)^3}\)
\(f''''(a) = -\frac{6}{(1 + a)^4}\)
\(f'''''(a) = \frac{24}{(1 + a)^5}\)
\(f''''''(a) = -\frac{120}{(1 + a)^6}\)

\(f(0) = 0\)
\(f'(0) = 1\)
\(f''(0) = -1\)
\(f'''(0) = 2\)
\(f''''(0) = -6\)
\(f'''''(0) = 24\)
\(f''''''(0) = -120\)

at a = 0:
\(f(x)= \frac{f(0)}{n!}(x−0)^n + \frac{f'(0)}{n!}(x−0)^n + \frac{f''(0)}{n!}(x−0)^n + \frac{f'''(0)}{n!}(x−0)^n + \frac{f''''(0)}{n!}(x−0)^n + \frac{f'''''(0)}{n!}(x−0)^n + \frac{f''''''(0)}{n!}(x−0)^n + ...\)
\(f(x)= \frac{0}{0!}(x)^0 + \frac{1}{1!}(x)^1 + \frac{-1}{2!}(x)^2 + \frac{2}{3!}(x)^3 + \frac{-6}{4!}(x)^4 + \frac{24}{5!}(x)^5 + \frac{-120}{6!}(x)^6 + ...\)
\(f(x)= 0 + x - \frac{x^2}{2!} + \frac{2x^3}{3!} - \frac{6x^4}{4!} + \frac{24x^5}{5!} - \frac{120x^6}{6!} + ...\)
\(f(x)= x - \frac{x^2}{2} + \frac{2x^3}{6} - \frac{6x^4}{24} + \frac{24x^5}{120} - \frac{120x^6}{720} + ...\)
The Taylor Series expansion of \(f(x) = ln(1 + x) = x - \frac{x^2}{2} + \frac{2x^3}{6} - \frac{6x^4}{24} + \frac{24x^5}{120} - \frac{120x^6}{720} + ...\)

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

\(f(a) = a^\frac{1}{2}\)
\(f'(a) = \frac{1}{2a^\frac{1}{2}}\)
\(f''(a) = -\frac{1}{4a^\frac{3}{2}}\)
\(f'''(a) = \frac{3}{8a^\frac{5}{2}}\)
\(f''''(a) = -\frac{15}{16a^\frac{7}{2}}\)
\(f'''''(a) = \frac{105}{32a^\frac{9}{2}}\)
\(f''''''(a) = -\frac{945}{64a^\frac{11}{2}}\)

\(f(0) = 0\)
\(f'(0) = 0\)
\(f''(0) = 0\)
\(f'''(0) = 0\)
\(f''''(0) = 0\)
\(f'''''(0) = 0\)
\(f''''''(0) = 0\)

at a = 0:
\(f(x)= \frac{f(0)}{n!}(x−0)^n + \frac{f'(0)}{n!}(x−0)^n + \frac{f''(0)}{n!}(x−0)^n + \frac{f'''(0)}{n!}(x−0)^n + \frac{f''''(0)}{n!}(x−0)^n + \frac{f'''''(0)}{n!}(x−0)^n + \frac{f''''''(0)}{n!}(x−0)^n + ...\)
\(f(x)= \frac{0}{0!}(x)^0 + \frac{0}{1!}(x)^1 + \frac{0}{2!}(x)^2 + \frac{0}{3!}(x)^3 + \frac{0}{4!}(x)^4 + \frac{0}{5!}(x)^5 + \frac{0}{6!}(x)^6 + ...\)
\(f(x)= 0x^0 + 0x^1 + 0x^2 + 0x^3 + 0x^4 + 0x^5 + 0x^6 + ...\)
\(f(x)= 0 + 0 + 0 + 0 + 0 + 0 + 0 + ...\)
The Taylor Series expansion of \(f(x)=x^\frac{1}{2} = 0 + 0 + 0 + 0 + 0 + 0 + 0 + ... = 0\)