I'm really struggling with trying to come up with the error bound when doing taylor series problems
Use the reaminder term to estimate the absolute error in approximating the following quantitites with the nth-order Taylor Polynomial cnetered at 0. Estimates are not unique.
If n is a fixed integer, suppose there exists a number M such that |f^(n+1)(c)|<=M for all c between a and x inclusive. The remainder in nth-order Taylor polynomial for f centered at a satisfies
The Attempt at a Solution
So apparently if we let f(x)=e^x, then f^(5)(x) = e^x, which I agree with fully
But I don't see how e^.25 is bounded by 2 and this is what we set M equal to. e^.25 is about 1.284025417...
I understood the previous similar problem that I did with sin(.3); n=4, because the nth derivative of sin(x) is either +/- sin(x) or +/- cos(x) which will always be <= 1 so this makes sense to use for a value of M, M=1 in this problem...
but for the one that I'm stuck on I don't see why we set M=2. The nth derivative of e^x is always going to be e^x which doesn't have a max value and increases without bound...
I hope it makes sense as to were I'm getting confused. Thanks for any help.