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White_Owl

Member

Registered: 2010-03-03

Posts: 106

Estimating a sum of a series

The problem: Calculate a partial sum for the first ten terms, estimate the error.  Round answers to five decimal places.

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:

And to solve this I need to find an integral of that scary function. But I have no idea how to approach this integral.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Estimating a sum of a series

Hi;

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.

Last edited by bobbym (2013-03-17 11:23:14)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

White_Owl

Member

Registered: 2010-03-03

Posts: 106

Re: Estimating a sum of a series

Could you please elaborate: what other means?
I am reading Stewart's Calculus Early Transcendentals (7th edition) and there is only one method described (ch 11.3)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Estimating a sum of a series

Hi;

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

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

White_Owl

Member

Registered: 2010-03-03

Posts: 106

Re: Estimating a sum of a series

Oh.... I see it know.
Actually the approach you used is discussed in calculus - the Squeeze Theorem. I just did not think it can be used here as well.
Thank you.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Estimating a sum of a series

Hi;

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.

Last edited by bobbym (2013-03-17 20:30:49)

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.