At this point, you might have to use the well-known inequality .
As for exact error, I don't think that it is easy to compute. It isn't a good use of your time.
I need to calculate the upper bound for the error (and the exact error) in the following approximation using Taylor Polynomials:
Where is the 5th degree Taylor Polynomial for
Taylors Theorem states that the error is where
This is where I'm lost, because is the max value on this interval. So is the upper bound for the error just
It just seems strange because e is the number I'm approximating. And how do I calculate exact 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.
(you are allowed to use a calculator to get the approximate value for this to 14 decimal places)
CBCode:(%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