Taylor Series

Question 1

Taylor series

Answer 1

Series [f^n(a)/n!] (x-a)^n f^n is the nth derivative

Question 2

MacLaurin

Answer 2

Taylor series centered at a=0

1. take the first several derivatives of the function
2. sub in a=0 for x in the function
3. Determine Cn based on Cn=f^n(a)/n!
4. Write out polynomial for series
5. Determine the general form

Question 3

MacLaurin for e^x

Answer 3

series [1/n!] (x)^n

If asked for MacLaurin series for e^3x2, the series is just series [1/n!] (3x^2)^n

Question 4

MacLaurin for sin(x)

Answer 4

Series [(-1)^n/(2n+1)!] (x)^2n+1

2n+1 is the general form for odd numbers

Question 5

MacLaurin for cos(x)

Answer 5

series [(-1)^n/(2n)!] (x)^2n

Question 6

Taylor polynomials

Answer 6

Tn(x)=f(a) + f’(a) (x-a) + f’‘(a)/2! (x-a)^2 + f’’‘(a)/3! (x-a)^3 +…+ f^n(a)/n! (x-a)^n

Could be asked for find fifth degree polynomial (n=5) or the first four non zero terms.