Originally Posted by
adkinsjr The problem was worded like this: Use Taylor's Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error.
I like the way you set the bounds for the error. That seems smart than just leaving the error with the "e" in it, because if we're calculating the error of an approximation for "e." Technically, we aren't supposed to know the value...
But your method helps with that dilemma, but I still don't see how to find the "exact value" of the error.
The exact value of the error is:
(you are allowed to use a calculator to get the approximate value for this to 14 decimal places)
Code:
(%i1) p5(x):=1+x+x^2/2!+x^3/3!+x^4/4!+x^5/5! /*define the Taylor polynomial*/;
(%o1) p5(x):=1+x+x^2/2!+x^3/3!+x^4/4!+x^5/5!
(%i2) p5(1) /*evaluate the approximation to e: p5(1)*/;
(%o2) 163/60
(%i3) (p5(1)/(1-1/6!))/6! /*compute the error bound*/;
(%o3) 163/43140
(%i4) float(%), numer /*convert the error bound to a float*/;
(%o4) 0.0037783959202596
(%i6) %e-p5(1) /*compute the actual error*/;
(%o6) %e-163/60
(%i7) float(%), numer /*convert actual error to float approximation*/;
(%o7) 0.0016151617923783
CB