How Can Divergent Integrals Be Handled with Ramanujan Summation?

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  • #1
Klaus_Hoffmann
86
1
Are there any method to deal with divergent integrals in the form

[tex] \int_{0}^{\infty}dx \frac{x^{3}}{x+1} [/tex] [tex] \int_{0}^{\infty}dx \frac{x}{(x+1)^{1/2}} [/tex] ?

in the same sense there are methods to give finite results to divergent series as 1+2+3+4+5+6+7+... or 1-4+9-16+25 or similar
 
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  • #2
WHAT methods give finite results to 1+2+3+4+5+6+7+... and 1-4+9-16+25 ?
 
  • #3
HallsofIvy said:
WHAT methods give finite results to 1+2+3+4+5+6+7+... and 1-4+9-16+25 ?

I've heard the term Ramanujan summation tossed about in regards to this, but I don't really know anything about it.
 

1. What is a divergent integral?

A divergent integral is an improper integral that does not have a finite value. This means that the integral does not converge to a specific number, but rather diverges to infinity or negative infinity.

2. How can you identify a divergent integral?

A divergent integral can be identified by the fact that it does not have a finite value. This can be seen by evaluating the integral and getting a result of infinity or negative infinity, or by using mathematical tests such as the comparison test or the limit comparison test.

3. What causes an integral to be divergent?

An integral can be divergent due to several reasons. One common cause is when the function being integrated has a vertical asymptote within the bounds of integration. Another cause is when the function grows too quickly, such as in the case of exponential functions.

4. Can a divergent integral be solved?

No, a divergent integral cannot be solved in the traditional sense. However, it may be possible to assign a finite value to the integral using a technique called analytic continuation. This involves extending the function being integrated beyond its original domain.

5. How are divergent integrals used in real-world applications?

Divergent integrals are commonly encountered in physics and engineering when dealing with infinite quantities, such as the electric field around a point charge or the gravitational force between two masses. They are also used in the study of infinite series and sequences in mathematics.

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