Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

Pages: **1**

**White_Owl****Member**- Registered: 2010-03-03
- Posts: 99

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.

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,892

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)*

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

Offline

**White_Owl****Member**- Registered: 2010-03-03
- Posts: 99

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)

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,892

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.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

Offline

**White_Owl****Member**- Registered: 2010-03-03
- Posts: 99

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.

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,892

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.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

Offline

Pages: **1**