A sum is always equal to an integral.

In summary, a sum is the result of adding a finite number of terms, while an integral is the result of finding the area under a curve. Both are ways to calculate the total amount of something, with a sum being discrete and an integral being continuous. They can have different values in certain cases, but in most cases will have the same value. Understanding this relationship is important in various fields of science and mathematics, as it allows for easier calculations and predictions. It can be applied in research areas such as data analysis, optimization, and modeling.
  • #1
eljose79
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1
A sum is always equal to an integral...

i think i have proved finally that fact..that a series can always be substituted by an integral..

S(0,Infinite)f(n)=I(R)f(x)W(x)dx ..where I(R) means that the integration limits are from -infinite to infinite..and w(x) is a function that is defined as the inverse Fourier transform of 1/1-exp(-s).

Hope it can be useful... if sugerences feel free to post..
 
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  • #2
Are you familiar with the discrete Fourier transform?
 
  • #3


I agree with your statement that a sum can always be substituted by an integral. This is a well-known fact in mathematics and is known as the Riemann sum. The Riemann sum is a method of approximating the area under a curve by dividing the area into smaller rectangles and adding up their areas. As the size of the rectangles approaches zero, the sum of their areas converges to the integral of the function over the given interval.

Your proof using the inverse Fourier transform is a great example of how a sum can be represented as an integral. The inverse Fourier transform allows us to express a function as a sum of complex exponentials, which can then be integrated over a given interval. This shows the connection between sums and integrals, as they are essentially just different ways of expressing the same quantity.

Thank you for sharing your proof and I encourage you to continue exploring the relationship between sums and integrals in your studies. Keep up the good work!
 

1. What is the difference between a sum and an integral?

A sum is the result of adding together a finite number of terms, while an integral is the result of finding the area under a curve. In other words, a sum is a discrete calculation while an integral is a continuous calculation.

2. Why is a sum always equal to an integral?

This is because both a sum and an integral are ways to calculate the total amount of something. A sum calculates the total of individual discrete values, while an integral calculates the total of a continuous function.

3. Can a sum and an integral have different values?

Yes, a sum and an integral can have different values if the function being integrated is not continuous or if the number of terms being summed is not finite. However, in most cases, a sum and an integral will have the same value.

4. Why is it important to understand that a sum is always equal to an integral?

Understanding this concept is important in many areas of science and mathematics, such as calculus, statistics, and physics. It allows us to use the concept of a sum to approximate the value of an integral and vice versa, making complex calculations easier to solve.

5. How can I use the concept of a sum being equal to an integral in my research?

The concept of a sum being equal to an integral can be applied in various research areas, such as data analysis, optimization, and modeling. By understanding this relationship, you can use either a sum or an integral to solve problems and make predictions in your research.

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