Editing Improper integral (section)

==Summability==
An improper integral may diverge in the sense that the limit defining it may not exist.  In this case, there are more sophisticated definitions of the limit which can produce a convergent value for the improper integral.  These are called [[summability]] methods.

One summability method, popular in [[Fourier analysis]], is that of [[Cesàro summation]].  The integral

:<math>\int_0^\infty f(x)\,dx</math>

is Cesàro summable (C,&nbsp;α) if

:<math>\lim_{\lambda\to\infty}\int_0^\lambda\left(1-\frac{x}{\lambda}\right)^\alpha f(x)\ dx</math>

exists and is finite {{harv|Titchmarsh|1948|loc=§1.15}}.  The value of this limit, should it exist, is the (C,&nbsp;α) sum of the integral.

An integral is (C,&nbsp;0) summable precisely when it exists as an improper integral.  However, there are integrals which are (C,&nbsp;α) summable for α&nbsp;>&nbsp;0 which fail to converge as improper integrals (in the sense of Riemann or Lebesgue). One example is the integral

:<math>\int_0^\infty\sin x \,dx</math>

which fails to exist as an improper integral, but is (C,''α'') summable for every ''α''&nbsp;>&nbsp;0.  This is an integral version of [[Grandi's series]].