# Summing up an Arithmetic Progression via Integration?

In summary, the conversation discusses the integration and summation of a general term of an arithmetic progression (A.P.). While the integral and sum are not equal, there is an inequality that can be used. However, this may not hold for all expressions, and for more complex expressions, Bernoulli polynomials may be used. There is also a clarification about a missing sign in the integral.
Why doesn't the integration of the general term of an A.P. give its sum? Integration sums up finctions, so if I integrate the general term function of an A.P., I should get its sum.
Like
2,4,6,8,...
T=2+(n-1)2=2n
$\int T dn$=n^2 ..(1)

Sum=S=(n/2)(4+(n-1)2)=(n/2)(2+2n)=n+(n^2) ..(2)

Why aren't these two equal?

I see no reason at all why the integral should equal the sum. The integral doesn't sum integers, it calculates area.

That said, we do have the following inequality (this does not hold in general!):

$$\sum_{k=0}^{n-1}{f(k)}\leq \int_0^n{f(x)dx}\leq\sum_{k=1}^n{f(k)}$$

This inequality is the best you can do, I fear...

micromass said:
I see no reason at all why the integral should equal the sum. The integral doesn't sum integers, it calculates area.

That said, we do have the following inequality (this does not hold in general!):

$$\sum_{k=0}^{n-1}{f(k)}\leq \int_0^n{f(x)dx}\leq\sum_{k=1}^n{f(k)}$$

This inequality is the best you can do, I fear...

If you are dealing with a polynomial expression, you can use what are called Bernoulli polynomials.

If the expression is not a simple one (as in some finite polynomial expression), then the inequality is a good bet, unless there are some tighter constraints for the specific expression.

Clarification:
Haven't you dropped a sign?
On the left hand integral, after integrating, I see
-ln|1 - x|

On the next or final line, the leading negative disappears.

## 1. What is an Arithmetic Progression?

An Arithmetic Progression is a sequence of numbers where the difference between any two consecutive terms is constant. For example, 2, 5, 8, 11 is an Arithmetic Progression where the difference between each term is 3.

## 2. What is Integration?

Integration is a mathematical process of finding the area under a curve. It is the inverse operation of differentiation and is used to find the original function from its derivative.

## 3. How can an Arithmetic Progression be summed up via Integration?

To sum up an Arithmetic Progression via Integration, we first need to express the progression in terms of a mathematical function. Then, we can use the integration formula to find the area under the curve and thus, the sum of the progression.

## 4. Why would we want to use Integration to sum up an Arithmetic Progression?

Integration provides a more accurate and efficient method for summing up an Arithmetic Progression. It allows us to find the sum of a progression without having to add up each individual term.

## 5. Can Integration be used for any type of Arithmetic Progression?

Yes, Integration can be used for any type of Arithmetic Progression, whether it is finite or infinite. However, the process for finding the sum may vary depending on the type of progression.

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