Discussion Prompt

Pick any exercise in 8.8 of the calculus textbook. Solve and post your solution. If you have issues doing so, discuss them.

Discussion Solution

Problem 8.8.8:

Find a formula for the nth term of the Taylor series of f(x),centered at c,by finding the coeffcients of the first few powers of x and looking for a pattern. (The formulas for several of these are found in Key Idea 32; show work verifying these formula.)

  • \(f(x)= 1/x;~c = 1\)

Solution:

We have:

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

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

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

So the nth term of the Taylor series for f(x) at c can be calculated as \((-1)^n * (x-1)^n\)