Question

Compute The Following Taylor Series To Third Order:

a F(X)=Cosx Near X=0
b f(X)=Lnx Near X=1

Answer 1

The third-order Taylor series for

`cos(x) near x=0 is cos(x) = 1 - x^2/2!. For ln(x) near x=1, it is ln(x) = (x-1) - (x-1)^2/2 + (x-1)^3/3.`

Step-by-step explanation:

To compute the Taylor series for f(x) = cos(x) near x=0, recall the Maclaurin series for cosine which is just a Taylor series centered at 0:

`cos(x) = 1 - x^2/2! + x^4/4! - ...`

To third order, we truncate after the

`x^2 term, since x^4`

is of the fourth order and beyond:

`cos(x) = 1 - x^2/2!`

Similarly, to find the Taylor series for f(x) = ln(x) near x=1 to third order, we can differentiate ln(x) to get the coefficients for the series:

• 
  `f(x) = ln(x)f'(x) = 1/xf''(x) = -1/x^2f'''(x) = 2/x^3`

Plugging x=1 into these derivatives we get the coefficients for the series:

`f(x) = f(1) + f'(1)*(x-1) + f''(1)*(x-1)^2/2! + f'''(1)*(x-1)^3/3! + ...The series for ln(x) near x=1 to third order is:ln(x) = (x-1) - (x-1)^2/2 + 2*(x-1)^3/6ln(x) = (x-1) - (x-1)^2/2 + (x-1)^3/3`