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Jhenrique
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If exist a connection between the infinitesimal derivative and the discrete derivative $$d = \log(\Delta + 1)$$ $$\Delta = \exp(d) - 1$$ exist too a coneection between summation ##\Sigma## and integration ##\int## ?
Stephen Tashi said:Knowing jhenrique's tastes, we should also mention: http://en.wikipedia.org/wiki/Euler–Maclaurin_formula
Jhenrique said:Extremely complicated! I already know this article, but yet so I oponed this thread with the hope that could exist a simple formula and somebody that knew it could post it here...
Summation and integration are two mathematical operations that involve adding up values. However, the main difference between them is that summation is used to add a finite number of values, while integration is used to add an infinite number of values. Summation is often represented by the symbol ∑ (sigma) and integration is represented by the symbol ∫ (integral).
Summation and integration are closely related to each other. In fact, integration is often referred to as a continuous version of summation. This is because integration involves finding the sum of infinitely small values, while summation involves finding the sum of a finite number of values. Both operations are used to find the total value of a function over a given range.
No, summation and integration cannot be used interchangeably. They may be related, but they are distinct operations with different purposes. Summation is used to find the total value of a function over a finite range, while integration is used to find the total value of a function over an infinite range. Additionally, summation is a discrete operation, while integration is a continuous operation.
Limits are an essential concept in both summation and integration. In summation, the limit is used to determine the range of values to be added up, while in integration, the limit is used to determine the upper and lower bounds of the integral. In both cases, the limit helps to define the range over which the operation will be performed.
The connection between summation and integration has many real-world applications, particularly in the fields of physics, engineering, and economics. For example, in physics, integration is used to calculate the work done by a force, which is the sum of the force over a given distance. In economics, integration is used to calculate the total revenue of a company, which is the sum of all sales over a given time period. In engineering, integration is used to calculate the total area under a curve, which is the sum of all the small rectangles that make up the curve.