Yeah. I'm on board with the 2nd derivative but that still doesn't help with the fact that

is not strictly decreasing in the range given since the 1st derivative is not negative at x=0. If it were negative @ x=0 then I think you've just finished it.
As far as the Taylor series goes, I think that you're just showing that the terms of the series are strictly decreasing in absolute value.
You need to prove something like
And I'm pretty sure that you can not remove the summation and compare the

terms so I'm not sure how you would do that.
This is a good discussion so be sure to add your input.
As an aside, are you using any application to help with some of your algebra? I have been looking for good free applications but I haven't been overly thrilled by what I've found yet and was hoping to hear your experiences.
As another aside, the first time that I viewed your newest post I didn't see all of the final set of equations and none of the attached image. Keep it in mind if it looks like I may have lost my mind at some point!