Find the Taylor series of the functions

Mathematics
Tutor: None Selected Time limit: 1 Day

Dec 12th, 2014

Talyor series centered at  Zo is T(Z) = f(Zo) + f'(Zo)(Z-Zo)^1/1! + f(Zo)(Z-Zo)^2/2! + ....  One way to do this is to take the derivative a few times, see if there's a patter, and create the series.

1. cosh z = [e^z + e^-z]/2  so f(z)= (e^2z +1)/2 ; f' = e^2z ; f''(z)= 2e^2z ; f''' = 2^2 e^2z etc. Find each of these at z=0, and we get f(n-prime) = 2^(n-1) .
So Taylor series is: Sum[n=0 to infinity]{(1/2)(2z)^(n)/n!}

2. f(z)= e^(z/3 - 2/3); f'= (1/3)e^(z/3 - 2/3) ; f''=(1/3)^2 e^(z/3 - 2/3) ect.
Series: e^(i/3-2/3)*Sum[n=0 to infinity]{(z-i)^n(1/3)^n}

3. f' = 12z² + 4z -1; f'' = 24z +4; f'''=24
Series: -6+7(z+1)/1! -20(z+1)²/2! +24(z+1)³/3! which should just be f(z).

Dec 13th, 2014

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Dec 12th, 2014
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