A Question on Integrating Expansions of Infinities

[eljose]

A question on infinities...

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

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

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

\frac{sqrt\pi}{2}=<br /> \sum_0^{\infty}\frac{a_n{\infty}^{n+1}}{n¡(n+1)}

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)

 

[matt grime]

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.)

 

[mathman]

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.