Taylor+and+McLaurin+Series

= = media type="custom" key="6245951" Many functions can be defined with a power series. For example, a function f(x) = 1/(1-x) can be estimated by using the power series for the function f(x). So for example, if you're trying to find f(3), to find an estimate for f(3) all you have to do is plug in 3 for x in the power series for f(x).

A MacLauren series is a series expansion of a function about the point x = 0



A Taylor series is a series expansion of a function about a point.




 * Step by Step Process for a MacLaurin Series:**

1. Determine what your degree of expansion is (ex. if the 3rd degree, it's out until the exponent is 3) 2. Find the value of your function at x = 0 3. Derive your function and find its value at x = 0 4. Find the 2nd derivative of your function and its value at x = 0 5. Keep deriving and plugging in zero until you reach the desired degree 6. Follow the pattern for the MacLaurin series as showed above - the first term is just your value at f(0), then your second term is the value of, then your third term is the value of , your third term is the value of , etc. 7. Now you have a MacLaurin series for your equation about x = 0

A Taylor series has a similar process to the MacLaurin series, however it's about a point a =/= 0
 * Step by Step Process for a Taylor Series:**

1. Determine what your degree of expansion is (ex. if the 3rd degree, it's out until the exponent is 3) 2. Find the value of your function at said point (x = a) 3. Derive your function and find its value at a 4. Find the 2nd derivative of your function and its value at a 5. Keep deriving and plugging in point a until you reach the desired degree 6. Follow the pattern for the Taylor series as showed above - the first term is just your value at f(a), then your second term is the value of, then your third term is the value of , your third term is the value of , etc. 7. Now you have a Taylor series for your equation about x = a


 * Examples of a MacLaurin Series:**

for a 2nd degree polynomial



So, plugging in those values into our MacLaurin series format

Which simplifies to...

for a 3rd degree polynomial



Now, we plug in those values into our MacLaurin series format

Which simplifies to...




 * Examples of a Taylor Series:**

for a 3rd degree polynomial about x = 1 (a = 1)



Plug in these values into the Taylor series format



for a 4th degree polynomial about



Substitute...