Find the Maclaurin series..

Question

The fuction sinh(x) is defined by sinh(X)= ( e^x - e^-x) / 2) 

 

Find the maclaurin series and use the first two terms to estimate sinh(1/2) . The series should be written using summation notation , the estimate should be given as a reduced fraction, and you do not need to provide an error bound for the estimate.

 

In my solutions booklet, it says that the even powered terms cancel. I dont understand why. It also says sinh(x)=1/2(e^x -e^-x )= x+ (x³/3!) + (x⁵ /5!)+... I dont understand why are these terms the answer.  

Answer 1

f(x) = sinhx = (e^x - e^-x)/2    f(0) = (e⁰ - e^-0)/2 = 0

 

f'(x) = (e^x + e^-x)/2              f'(0) = 1

 

f"(x) = (e^x - e^-x)/2              f"(0) = 0

 

f'''(x) = (e^x + e^-x)/2            f'''(0) = 1

 

MacLaurin Series for f(x):    f(0) + f'(0)(x-0) + f"(0)/2! (x-0)² + f'''(0)/3! (x-0)³ + ...

 

Substituting into the formula, sinhx = 0 + x + 0 + x³/3! + 0 + x⁵/5! + ...

 

                                                    = x + x³/3! + x⁵/5! + x⁷/7! + ...

 

Using the first 2 nonzero terms, sinh(1/2) ≈ (1/2) + (1/2)³/3!

 

                                                             = 1/2 + 1/48 = 0.520833333

 

Note:  Using a calculator, sinh(1/2) = (e^½ - e^-1/2)/2 ≈ 0.521095