Borel’s Integral and ^-Series. 
385 
1909-10.] 
the integral 
“ %x)d{qx) 
0 
the reader can easily construct a proof of convergency on the lines of the 
usual proof for 
\ 
limit f X n e~ x dx . 
a> / 
J 0 
Just as Bromwich deduces Euler’s series from Borel’s integral, so, on parallel 
lines, it is easy to deduce the modified form of Euler’s transformation which 
is given at the beginning (4) of this note. This may be left as an exercise 
for the reader. 
{Issued separately April 20, 1910.) 
