Assignment 14

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

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 a R-Markdown document.

Solutions

1. \(f(x) = 1/(1 - x)\)

We have:

  • first term: \(f(0) = 1\)
  • second term: \(f'(0)/1!*(x)^1 = -1*(1-0)^{-2} * -1/1! * x = x\)
  • third term: \(f''(0)/2!*(x)^2 = -2*1^{-3}/-2*(x)^2 = x^2\)
  • fourth term: \(f'''(0)/3!*(x)^3 = -6*1^{-4}/6*(x)^3 = x^3\)

So the Taylor series for \(f(x) = 1/(1 - x)\) is:

  • \(1 + x + x^2 +x^3 + ...\)
  • \(=\sum_{n=0}^\inf x^n\)

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

We have:

  • first term: \(f(0) = 1\)
  • second term: \(f'(0)/1!*(x)^1 = 1* x = x\)
  • third term: \(f''(0)/2!*(x)^2 = 1/2*(x)^2 = x^2/2\)
  • fourth term: \(f'''(0)/3!*(x)^3 = 1/6*(x)^3 = x^3/6\)

So the Taylor series for \(f(x) = e^x\) is:

  • \(1 + x + x^2/2! +x^3/3! + ...\)
  • \(=\sum_{n=0}^\inf x^n/n!\)

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

We have:

  • first term: \(f(0) = 0\)
  • second term: \(f'(0)/1!*(x)^1 = 1/(1+0)* x = x\)
  • third term: \(f''(0)/2!*(x)^2 = -1*1^{-2}/2*(x)^2 = -x^2/2\)
  • fourth term: \(f'''(0)/3!*(x)^3 = 2*1^{-3}/6*(x)^3 = x^3/3\)

So the Taylor series for \(f(x) =ln(1+x)\) is:

  • \(0 + x - x^2/2 +x^3/3 + ...\)
  • \(=\sum_{n=1}^\inf (-1)^{n+1}*x^n/n\)