A Question on Integrating Expansions of Infinities

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Therefore, the original integral is sqrt(pi)/2.In summary, the conversation discusses the question of whether a sum of infinities can result in a finite number, particularly in the case of evaluating the integral of the function e^{-x^2}. It is concluded that the simplest approach to evaluating this integral is to square it and change the variables to polar coordinates, resulting in a final answer of sqrt(pi)/2.
  • #1
eljose
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A question on infinities...

we all know that the function [tex]e^{-x^2}[/tex] can be expanded into a taylor series my question arises when we try to perform the integral:

[tex]\int_0^{\infty}exp(-x^2)=\frac{sqrt\pi}{2}[/tex]

then if we expand exp(-x^2) in terms of its Taylor series and perform the integration we would find that:

[tex]\frac{sqrt\pi}{2}=
\sum_0^{\infty}\frac{a_n{\infty}^{n+1}}{n¡(n+1)}[/tex]

the question is if a sum of infinities can give a finite number such as happens in the last sum... where the a_n are the taylor coefficients of the series expansion for exp(-x^2)
 
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  • #2
The sum makes no sense: you're not even adding up real numbers. At least take limits first, and then ask if it is permissible to interchange limits and summation signs. (answer, not always - this is basic analysis, not number theory.)
 
  • #3
To evaluate the integral, the simplest approach that works is first square it, then change the variables in the double integral to polar coordinates. You will then very easily get pi/4 for the squared integral.
 

What is infinity?

Infinity is a concept that refers to something that has no limit or end. It is often used to describe something that is endless or boundless.

Can infinity be measured?

No, infinity cannot be measured because it is not a tangible quantity. It is a concept that represents something that is limitless.

Is infinity a number?

No, infinity is not considered a number in mathematics. It is a concept that is used to represent something that is unbounded.

Can infinity be divided?

No, infinity cannot be divided. Division is a mathematical operation that requires a finite quantity to work with, and infinity is not a finite quantity.

Are there different types of infinity?

Yes, there are different types of infinity in mathematics. For example, there is countable infinity, which is used to describe infinite sets that can be put into a one-to-one correspondence with the natural numbers. There is also uncountable infinity, which is used to describe infinite sets that cannot be put into a one-to-one correspondence with the natural numbers.

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