SUMMING A DIVERGENT SERIES
D. G. C. MKeon
It is well known that the manipulation of divergent series an give
1
X
peuliar results. The series S (a) =
an n! is learly horribly
n=0
divergent for a 6= 0, yet we will show by using some elementary
integrals involving the Bessel funtion K0 (x) that it an be viewed
as an expansion in powers of a of a funtion whih diverges for
a < 0.
The two integrals we need are [1℄
Z 1
p 1
(1)
dt tn K0 (2 t) = (n!)2
2
0
and
f (a) = 2
Z 1
dt e
at
p
K (2 t) =
0
0
1
ea
a
Z 1
1
=a
dt
e
t
t
(a > 0):
(2)
Let us onsider expanding e at in the argument of (2) so that
Z 1 X
1 ( a) n
p
(3)
tn K0 (2 t);
f (a) = 2
dt
n!
0
n=0
integration of this term by term using (1) yields
1 ( a) n 1
X
2
(n!)
f (a) = 2
n!
2
n=0
=
1
X
n=0
( a)n n!
45
(4)
IMS Bulletin 42, 1999
46
Thus the divergent series S ( a) an be onsidered as a representation of the funtion f (a), whih from its denition in (2), is
learly singular for a < 0! (We note that f (a) ! 1 as a ! 0+ in
both (2) and (4).) This may alternatively be viewed as a way of
inserting a \onvergene" fator of 1=(n!)2 into eah term of the
series for S ( a).
These manipulations are akin to the insertion of a fator of
1
in
Borel
series. To see this, onsider the integrals
n!
Z 1
dt tn e
t
= n!
(5)
1
(a > 1):
1+a
(6)
0
and
g (a) =
Z 1
at
dt e
e
0
Expanding e
at
t
=
in powers of at in (6) we obtain
g (a) =
Z 1
dt
0
1 ( at)n
X
e
n!
n=0
t
(7)
whih beomes, if we integrate term-by-term using (5)
g (a) =
=
1 ( a) n
X
(n!)
n!
n=0
1
X
n=0
(8)
( a)n :
1
is formally represBy omparing (6) with (8), we see that
1+a
1
X
( a)n for all a > 1 even though this
ented by the series
n=0
geometri series diverges for jaj > 1.
We hene see that, one again, a divergent series whih at
rst glane is \meaningless" may in fat be given a \meaning",
Summing a divergent series
47
provided we are willing to indulge in interhanging the order of
summation and integration, as we have done in going from (2) to
1 ( at)n
X
uniformly onverges to e at for
(4) or (6) to (8). (As
n
!
n=0
nite at, term-by-term integration of the series in (3) and (7) is
justied over any nite range.)
A more formal approah that is learly related to the usual
disussion of Borel Summation is now skethed. If one has a series
1
X
f (a) =
bn an n!
(9)
n=0
whih onverges for jaj < R, then
(a) =
1 b an
X
n
n!
n=0
(10)
onverges everywhere. It is easily shown now, using the integral
of eq. (1), that
Z 1
p
(11)
F (a) = 2
dt (ta)K0 (2 t)
0
is an analyti ontinuation of f (a) aross jaj = R wherever F (a)
is analyti. We have essentially onsidered an example of this
general result for the ase bn = ( 1)n and R = 0 where the usual
arguments are invalid.
Referenes
[1℄
I. Gradsteyn and M. Ryzhik, Table of Integrals, Series and Produts,
5th ed. Aademi Press.
D. C. G. MKeon
Department of Applied Mathematis
University of Western Ontario
London
Canada N6A 5B7