You can use that to estimate the tail more easily. But in this case it will not be as sharp a bound as using a numerical idea on the Taylor form of the remainder.
]]>This is a problem that comes under the heading of Numerical Analysis and is never discussed in calculus or analysis. They are more concerned with whether a series converges or not. They are not concerned with what it converges to.
The simplest bounding method is to look at the integrand and see that
]]>There are other ways to bound that sum that do not require the evaluation of the Taylor form of the remainder. Using one you can immediately get:
Would they suffice?
The integral has a known closed form but not in terms of elementary functions. We could numerically integrate it or as I said we could use other means to bound the tail.
]]>Well, the partial sum is not a problem - a few minutes with a calculator and the answer is 1.04931. But the error estimation part gives me the trouble.
As far as I understand, I am supposed to solve: