Taylorseries

Slide 1: Taylor/ Maclaurin Series

Slide 2: Taylor series  A Taylor series is an infinite sum that is based on a parent function. To create a Taylor series from any equation- use to general term: ﾥ f (a)( x - a ) n n ￥ n! n =0 In the general term, a is any value on the function that you use to calculate the derivatives, in Maclaurin series, a=0.

Slide 3: Examples  Remember the general term  Discover patterns in derivatives that will make the solution easier. ﾥ f (a )( x - a ) n n ￥ n! n=0

Slide 4: Try a Maclaurin Series with f ( x) = e x Step 1: find the derivatives for: f (a) = e a Remember that this particular f '(a ) = e a function is easy because the f ''(a ) = e a pattern is easy to spot. f '''(a ) = e a ...

Slide 5: Step 2, use a=0 and find your coefficient (it’s a maclaurin series) f (a) = e a f (0) = e = 1 0 f '(a ) = e a f '(0) = e = 1 0 f ''(a ) = e a f ''(0) = e = 1 0 f '''(a ) = e a f '''(0) = e = 1 0 ... ...

Slide 6: Now create your Maclaurin series  Remember a=0 1( x - 0) 2( x - 0) 3( x - 0) 1 2 3 e = x + + 1! 2! 3! 4( x - 0) 4 n( x - 0) n + + ... + + ... 4! n!

Slide 7: The Maclaurin series to M ORI ZE EM 1 x x2 x3 x4 x5 ex = + + + + + ... 0! 1! 2! 3! 4! 5! x x3 x5 x7 x9 sin( x ) = - + - + ... 1! 3! 5! 7! 9! -1 x x3 x5 x7 x9 tan ( x ) = - + - + ... 1 3 5 7 9 1 x2 x4 x6 x8 cos( x ) = - + - + ... 0! 2! 4! 6! 8! x x2 x3 x4 x5 ln( x + 1) = - + - + ... 1 2 3 4 5 1 = 1 + x + x 2 + x 3 + x 4 + x 5 ... 1- x