<s>
In	O
the	O
mathematics	O
of	O
convergent	B-Algorithm
and	O
divergent	B-Algorithm
series	I-Algorithm
,	O
Euler	B-Algorithm
summation	I-Algorithm
is	O
a	O
summation	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
As	O
well	O
as	O
being	O
used	O
to	O
define	O
values	O
for	O
divergent	B-Algorithm
series	I-Algorithm
,	O
Euler	B-Algorithm
summation	I-Algorithm
can	O
be	O
used	O
to	O
speed	O
the	O
convergence	B-Algorithm
of	O
series	O
.	O
</s>
<s>
Euler	B-Algorithm
summation	I-Algorithm
can	O
be	O
generalized	O
into	O
a	O
family	O
of	O
methods	O
denoted	O
(	O
E	O
,	O
q	O
)	O
,	O
where	O
q	O
≥	O
0	O
.	O
</s>
<s>
However	O
,	O
using	O
Euler	B-Algorithm
summation	I-Algorithm
can	O
accelerate	B-Algorithm
the	I-Algorithm
convergence	I-Algorithm
(	O
this	O
is	O
especially	O
useful	O
for	O
alternating	O
series	O
)	O
;	O
sometimes	O
it	O
can	O
also	O
give	O
a	O
useful	O
meaning	O
to	O
divergent	O
sums	O
.	O
</s>
<s>
Using	O
y	O
=	O
1	O
for	O
the	O
formal	O
sum	O
we	O
get	O
if	O
Pk	O
is	O
a	O
polynomial	O
of	O
degree	O
k	O
.	O
Note	O
that	O
the	O
inner	O
sum	O
would	O
be	O
zero	O
for	O
,	O
so	O
in	O
this	O
case	O
Euler	B-Algorithm
summation	I-Algorithm
reduces	O
an	O
infinite	O
series	O
to	O
a	O
finite	O
sum	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
formal	O
sum	O
in	O
this	O
case	O
diverges	O
since	O
k	O
is	O
positive	O
,	O
but	O
applying	O
Euler	B-Algorithm
summation	I-Algorithm
to	O
the	O
zeta	O
function	O
(	O
or	O
rather	O
,	O
to	O
the	O
related	O
Dirichlet	O
eta	O
function	O
)	O
yields	O
(	O
cf	O
.	O
</s>
<s>
Globally	O
convergent	B-Algorithm
series	I-Algorithm
)	O
which	O
is	O
of	O
closed	O
form	O
.	O
</s>
