<s>
In	O
mathematics	O
,	O
a	O
divergent	B-Algorithm
series	I-Algorithm
is	O
an	O
infinite	O
series	O
that	O
is	O
not	O
convergent	B-Algorithm
,	O
meaning	O
that	O
the	O
infinite	O
sequence	O
of	O
the	O
partial	O
sums	O
of	O
the	O
series	O
does	O
not	O
have	O
a	O
finite	O
limit	B-Algorithm
.	O
</s>
<s>
However	O
,	O
convergence	B-Algorithm
is	O
a	O
stronger	O
condition	O
:	O
not	O
all	O
series	O
whose	O
terms	O
approach	O
zero	O
converge	O
.	O
</s>
<s>
The	O
divergence	B-Algorithm
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
was	O
proven	O
by	O
the	O
medieval	O
mathematician	O
Nicole	O
Oresme	O
.	O
</s>
<s>
In	O
specialized	O
mathematical	O
contexts	O
,	O
values	O
can	O
be	O
objectively	O
assigned	O
to	O
certain	O
series	O
whose	O
sequences	O
of	O
partial	O
sums	O
diverge	O
,	O
in	O
order	O
to	O
make	O
meaning	O
of	O
the	O
divergence	B-Algorithm
of	O
the	O
series	O
.	O
</s>
<s>
A	O
summability	B-Algorithm
method	I-Algorithm
or	O
summation	B-Algorithm
method	I-Algorithm
is	O
a	O
partial	B-Algorithm
function	I-Algorithm
from	O
the	O
set	O
of	O
series	O
to	O
values	O
.	O
</s>
<s>
Cesàro	B-Algorithm
summation	I-Algorithm
is	O
an	O
averaging	O
method	O
,	O
in	O
that	O
it	O
relies	O
on	O
the	O
arithmetic	O
mean	O
of	O
the	O
sequence	O
of	O
partial	O
sums	O
.	O
</s>
<s>
Other	O
methods	O
involve	O
analytic	B-Algorithm
continuations	I-Algorithm
of	O
related	O
series	O
.	O
</s>
<s>
In	O
physics	O
,	O
there	O
are	O
a	O
wide	O
variety	O
of	O
summability	B-Algorithm
methods	I-Algorithm
;	O
these	O
are	O
discussed	O
in	O
greater	O
detail	O
in	O
the	O
article	O
on	O
regularization	O
.	O
</s>
<s>
Before	O
the	O
19th	O
century	O
,	O
divergent	B-Algorithm
series	I-Algorithm
were	O
widely	O
used	O
by	O
Leonhard	O
Euler	O
and	O
others	O
,	O
but	O
often	O
led	O
to	O
confusing	O
and	O
contradictory	O
results	O
.	O
</s>
<s>
A	O
major	O
problem	O
was	O
Euler	O
's	O
idea	O
that	O
any	O
divergent	B-Algorithm
series	I-Algorithm
should	O
have	O
a	O
natural	O
sum	O
,	O
without	O
first	O
defining	O
what	O
is	O
meant	O
by	O
the	O
sum	O
of	O
a	O
divergent	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
Augustin-Louis	O
Cauchy	O
eventually	O
gave	O
a	O
rigorous	O
definition	O
of	O
the	O
sum	O
of	O
a	O
(	O
convergent	B-Algorithm
)	O
series	O
,	O
and	O
for	O
some	O
time	O
after	O
this	O
,	O
divergent	B-Algorithm
series	I-Algorithm
were	O
mostly	O
excluded	O
from	O
mathematics	O
.	O
</s>
<s>
In	O
1890	O
,	O
Ernesto	O
Cesàro	B-Algorithm
realized	O
that	O
one	O
could	O
give	O
a	O
rigorous	O
definition	O
of	O
the	O
sum	O
of	O
some	O
divergent	B-Algorithm
series	I-Algorithm
,	O
and	O
defined	O
Cesàro	B-Algorithm
summation	I-Algorithm
.	O
</s>
<s>
(	O
This	O
was	O
not	O
the	O
first	O
use	O
of	O
Cesàro	B-Algorithm
summation	I-Algorithm
,	O
which	O
was	O
used	O
implicitly	O
by	O
Ferdinand	O
Georg	O
Frobenius	O
in	O
1880	O
;	O
Cesàro	B-Algorithm
's	O
key	O
contribution	O
was	O
not	O
the	O
discovery	O
of	O
this	O
method	O
,	O
but	O
his	O
idea	O
that	O
one	O
should	O
give	O
an	O
explicit	O
definition	O
of	O
the	O
sum	O
of	O
a	O
divergent	B-Algorithm
series	I-Algorithm
.	O
)	O
</s>
<s>
In	O
the	O
years	O
after	O
Cesàro	B-Algorithm
's	O
paper	O
,	O
several	O
other	O
mathematicians	O
gave	O
other	O
definitions	O
of	O
the	O
sum	O
of	O
a	O
divergent	B-Algorithm
series	I-Algorithm
,	O
although	O
these	O
are	O
not	O
always	O
compatible	O
:	O
different	O
definitions	O
can	O
give	O
different	O
answers	O
for	O
the	O
sum	O
of	O
the	O
same	O
divergent	B-Algorithm
series	I-Algorithm
;	O
so	O
,	O
when	O
talking	O
about	O
the	O
sum	O
of	O
a	O
divergent	B-Algorithm
series	I-Algorithm
,	O
it	O
is	O
necessary	O
to	O
specify	O
which	O
summation	B-Algorithm
method	I-Algorithm
one	O
is	O
using	O
.	O
</s>
<s>
A	O
summability	B-Algorithm
method	I-Algorithm
M	O
is	O
regular	O
if	O
it	O
agrees	O
with	O
the	O
actual	O
limit	B-Algorithm
on	O
all	O
convergent	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
Here	O
partial	O
converse	O
means	O
that	O
if	O
M	O
sums	O
the	O
series	O
Σ	O
,	O
and	O
some	O
side-condition	O
holds	O
,	O
then	O
Σ	O
was	O
convergent	B-Algorithm
in	O
the	O
first	O
place	O
;	O
without	O
any	O
side-condition	O
such	O
a	O
result	O
would	O
say	O
that	O
M	O
only	O
summed	O
convergent	B-Algorithm
series	I-Algorithm
(	O
making	O
it	O
useless	O
as	O
a	O
summation	B-Algorithm
method	I-Algorithm
for	O
divergent	B-Algorithm
series	I-Algorithm
)	O
.	O
</s>
<s>
The	O
function	O
giving	O
the	O
sum	O
of	O
a	O
convergent	B-Algorithm
series	I-Algorithm
is	O
linear	O
,	O
and	O
it	O
follows	O
from	O
the	O
Hahn	O
–	O
Banach	O
theorem	O
that	O
it	O
may	O
be	O
extended	O
to	O
a	O
summation	B-Algorithm
method	I-Algorithm
summing	O
any	O
series	O
with	O
bounded	O
partial	O
sums	O
.	O
</s>
<s>
This	O
is	O
called	O
the	O
Banach	B-Algorithm
limit	I-Algorithm
.	O
</s>
<s>
The	O
subject	O
of	O
divergent	B-Algorithm
series	I-Algorithm
,	O
as	O
a	O
domain	O
of	O
mathematical	O
analysis	O
,	O
is	O
primarily	O
concerned	O
with	O
explicit	O
and	O
natural	O
techniques	O
such	O
as	O
Abel	O
summation	O
,	O
Cesàro	B-Algorithm
summation	I-Algorithm
and	O
Borel	O
summation	O
,	O
and	O
their	O
relationships	O
.	O
</s>
<s>
Summation	O
of	O
divergent	B-Algorithm
series	I-Algorithm
is	O
also	O
related	O
to	O
extrapolation	O
methods	O
and	O
sequence	O
transformations	O
as	O
numerical	O
techniques	O
.	O
</s>
<s>
Summation	B-Algorithm
methods	I-Algorithm
usually	O
concentrate	O
on	O
the	O
sequence	O
of	O
partial	O
sums	O
of	O
the	O
series	O
.	O
</s>
<s>
While	O
this	O
sequence	O
does	O
not	O
converge	O
,	O
we	O
may	O
often	O
find	O
that	O
when	O
we	O
take	O
an	O
average	O
of	O
larger	O
and	O
larger	O
numbers	O
of	O
initial	O
terms	O
of	O
the	O
sequence	O
,	O
the	O
average	O
converges	O
,	O
and	O
we	O
can	O
use	O
this	O
average	O
instead	O
of	O
a	O
limit	B-Algorithm
to	O
evaluate	O
the	O
sum	O
of	O
the	O
series	O
.	O
</s>
<s>
A	O
summation	B-Algorithm
method	I-Algorithm
can	O
be	O
seen	O
as	O
a	O
function	O
from	O
a	O
set	O
of	O
sequences	O
of	O
partial	O
sums	O
to	O
values	O
.	O
</s>
<s>
If	O
A	O
is	O
any	O
summation	B-Algorithm
method	I-Algorithm
assigning	O
values	O
to	O
a	O
set	O
of	O
sequences	O
,	O
we	O
may	O
mechanically	O
translate	O
this	O
to	O
a	O
series-summation	O
method	O
AΣ	O
that	O
assigns	O
the	O
same	O
values	O
to	O
the	O
corresponding	O
series	O
.	O
</s>
<s>
If	O
a	O
and	O
a′	O
are	O
two	O
series	O
such	O
that	O
there	O
exists	O
a	O
bijection	B-Algorithm
such	O
that	O
for	O
all	O
i	O
,	O
and	O
if	O
there	O
exists	O
some	O
such	O
that	O
for	O
all	O
i>N	O
,	O
then	O
(	O
In	O
other	O
words	O
,	O
a′	O
is	O
the	O
same	O
series	O
as	O
a	O
,	O
with	O
only	O
finitely	O
many	O
terms	O
re-indexed	O
.	O
)	O
</s>
<s>
This	O
is	O
a	O
weaker	O
condition	O
than	O
stability	O
,	O
because	O
any	O
summation	B-Algorithm
method	I-Algorithm
that	O
exhibits	O
stability	O
also	O
exhibits	O
finite	O
re-indexability	O
,	O
but	O
the	O
converse	O
is	O
not	O
true	O
.	O
)	O
</s>
<s>
A	O
desirable	O
property	O
for	O
two	O
distinct	O
summation	B-Algorithm
methods	I-Algorithm
A	O
and	O
B	O
to	O
share	O
is	O
consistency	O
:	O
A	O
and	O
B	O
are	O
consistent	O
if	O
for	O
every	O
sequence	O
s	O
to	O
which	O
both	O
assign	O
a	O
value	O
,	O
(	O
Using	O
this	O
language	O
,	O
a	O
summation	B-Algorithm
method	I-Algorithm
A	O
is	O
regular	O
iff	O
it	O
is	O
consistent	O
with	O
the	O
standard	O
sum	O
Σ	O
.	O
)	O
</s>
<s>
There	O
are	O
powerful	O
numerical	O
summation	B-Algorithm
methods	I-Algorithm
that	O
are	O
neither	O
regular	O
nor	O
linear	O
,	O
for	O
instance	O
nonlinear	O
sequence	O
transformations	O
like	O
Levin-type	O
sequence	O
transformations	O
and	O
Padé	O
approximants	O
,	O
as	O
well	O
as	O
the	O
order-dependent	O
mappings	O
of	O
perturbative	O
series	O
based	O
on	O
renormalization	O
techniques	O
.	O
</s>
<s>
Taking	O
regularity	O
,	O
linearity	O
and	O
stability	O
as	O
axioms	O
,	O
it	O
is	O
possible	O
to	O
sum	O
many	O
divergent	B-Algorithm
series	I-Algorithm
by	O
elementary	O
algebraic	O
manipulations	O
.	O
</s>
<s>
This	O
partly	O
explains	O
why	O
many	O
different	O
summation	B-Algorithm
methods	I-Algorithm
give	O
the	O
same	O
answer	O
for	O
certain	O
series	O
.	O
</s>
<s>
can	O
be	O
evaluated	O
regardless	O
of	O
convergence	B-Algorithm
.	O
</s>
<s>
More	O
rigorously	O
,	O
any	O
summation	B-Algorithm
method	I-Algorithm
that	O
possesses	O
these	O
properties	O
and	O
which	O
assigns	O
a	O
finite	O
value	O
to	O
the	O
geometric	B-Algorithm
series	I-Algorithm
must	O
assign	O
this	O
value	O
.	O
</s>
<s>
However	O
,	O
when	O
r	O
is	O
a	O
real	O
number	O
larger	O
than	O
1	O
,	O
the	O
partial	O
sums	O
increase	O
without	O
bound	O
,	O
and	O
averaging	O
methods	O
assign	O
a	O
limit	B-Algorithm
of	O
infinity	O
.	O
</s>
<s>
The	O
two	O
classical	O
summation	B-Algorithm
methods	I-Algorithm
for	O
series	O
,	O
ordinary	O
convergence	B-Algorithm
and	O
absolute	O
convergence	B-Algorithm
,	O
define	O
the	O
sum	O
as	O
a	O
limit	B-Algorithm
of	O
certain	O
partial	O
sums	O
.	O
</s>
<s>
These	O
are	O
included	O
only	O
for	O
completeness	O
;	O
strictly	O
speaking	O
they	O
are	O
not	O
true	O
summation	B-Algorithm
methods	I-Algorithm
for	O
divergent	B-Algorithm
series	I-Algorithm
since	O
,	O
by	O
definition	O
,	O
a	O
series	O
is	O
divergent	O
only	O
if	O
these	O
methods	O
do	O
not	O
work	O
.	O
</s>
<s>
Most	O
but	O
not	O
all	O
summation	B-Algorithm
methods	I-Algorithm
for	O
divergent	B-Algorithm
series	I-Algorithm
extend	O
these	O
methods	O
to	O
a	O
larger	O
class	O
of	O
sequences	O
.	O
</s>
<s>
Absolute	O
convergence	B-Algorithm
defines	O
the	O
sum	O
of	O
a	O
sequence	O
(	O
or	O
set	O
)	O
of	O
numbers	O
to	O
be	O
the	O
limit	B-Algorithm
of	O
the	O
net	O
of	O
all	O
partial	O
sums	O
,	O
if	O
it	O
exists	O
.	O
</s>
<s>
It	O
does	O
not	O
depend	O
on	O
the	O
order	O
of	O
the	O
elements	O
of	O
the	O
sequence	O
,	O
and	O
a	O
classical	O
theorem	O
says	O
that	O
a	O
sequence	O
is	O
absolutely	O
convergent	B-Algorithm
if	O
and	O
only	O
if	O
the	O
sequence	O
of	O
absolute	O
values	O
is	O
convergent	B-Algorithm
in	O
the	O
standard	O
sense	O
.	O
</s>
<s>
Cauchy	O
's	O
classical	O
definition	O
of	O
the	O
sum	O
of	O
a	O
series	O
defines	O
the	O
sum	O
to	O
be	O
the	O
limit	B-Algorithm
of	O
the	O
sequence	O
of	O
partial	O
sums	O
.	O
</s>
<s>
This	O
is	O
the	O
default	O
definition	O
of	O
convergence	B-Algorithm
of	O
a	O
sequence	O
.	O
</s>
<s>
then	O
the	O
limit	B-Algorithm
of	O
tn	O
as	O
n	O
goes	O
to	O
infinity	O
is	O
an	O
average	O
called	O
the	O
Nørlund	O
mean	O
Np(s )	O
.	O
</s>
<s>
The	O
most	O
significant	O
of	O
the	O
Nørlund	O
means	O
are	O
the	O
Cesàro	B-Algorithm
sums	I-Algorithm
.	O
</s>
<s>
then	O
the	O
Cesàro	B-Algorithm
sum	I-Algorithm
Ck	O
is	O
defined	O
by	O
Cesàro	B-Algorithm
sums	I-Algorithm
are	O
Nørlund	O
means	O
if	O
,	O
and	O
hence	O
are	O
regular	O
,	O
linear	O
,	O
stable	O
,	O
and	O
consistent	O
.	O
</s>
<s>
C0	O
is	O
ordinary	O
summation	O
,	O
and	O
C1	O
is	O
ordinary	O
Cesàro	B-Algorithm
summation	I-Algorithm
.	O
</s>
<s>
Cesàro	B-Algorithm
sums	I-Algorithm
have	O
the	O
property	O
that	O
if	O
then	O
Ch	O
is	O
stronger	O
than	O
Ck	O
.	O
</s>
<s>
More	O
generally	O
,	O
if	O
the	O
series	O
for	O
f	O
only	O
converges	O
for	O
large	O
x	O
but	O
can	O
be	O
analytically	B-Algorithm
continued	I-Algorithm
to	O
all	O
positive	O
real	O
x	O
,	O
then	O
one	O
can	O
still	O
define	O
the	O
sum	O
of	O
the	O
divergent	B-Algorithm
series	I-Algorithm
by	O
the	O
limit	B-Algorithm
above	O
.	O
</s>
<s>
Abelian	B-Algorithm
means	I-Algorithm
are	O
regular	O
and	O
linear	O
,	O
but	O
not	O
stable	O
and	O
not	O
always	O
consistent	O
between	O
different	O
choices	O
of	O
λ	O
.	O
</s>
<s>
However	O
,	O
some	O
special	O
cases	O
are	O
very	O
important	O
summation	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Abel	O
summation	O
is	O
interesting	O
in	O
part	O
because	O
it	O
is	O
consistent	O
with	O
but	O
more	O
powerful	O
than	O
Cesàro	B-Algorithm
summation	I-Algorithm
:	O
whenever	O
the	O
latter	O
is	O
defined	O
.	O
</s>
<s>
The	O
Abel	O
sum	O
is	O
therefore	O
regular	O
,	O
linear	O
,	O
stable	O
,	O
and	O
consistent	O
with	O
Cesàro	B-Algorithm
summation	I-Algorithm
.	O
</s>
<s>
Then	O
L(s )	O
,	O
the	O
Lindelöf	O
sum	O
,	O
is	O
the	O
limit	B-Algorithm
of	O
f(x )	O
as	O
x	O
goes	O
to	O
positive	O
zero	O
.	O
</s>
<s>
If	O
g(z )	O
is	O
analytic	O
in	O
a	O
disk	O
around	O
zero	O
,	O
and	O
hence	O
has	O
a	O
Maclaurin	O
series	O
G(z )	O
with	O
a	O
positive	O
radius	O
of	O
convergence	B-Algorithm
,	O
then	O
in	O
the	O
Mittag-Leffler	O
star	O
.	O
</s>
<s>
Moreover	O
,	O
convergence	B-Algorithm
to	O
g(z )	O
is	O
uniform	O
on	O
compact	O
subsets	O
of	O
the	O
star	O
.	O
</s>
<s>
Several	O
summation	B-Algorithm
methods	I-Algorithm
involve	O
taking	O
the	O
value	O
of	O
an	O
analytic	B-Algorithm
continuation	I-Algorithm
of	O
a	O
function	O
.	O
</s>
<s>
If	O
Σanxn	O
converges	O
for	O
small	O
complex	O
x	O
and	O
can	O
be	O
analytically	B-Algorithm
continued	I-Algorithm
along	O
some	O
path	O
from	O
x	O
=	O
0	O
to	O
the	O
point	O
x	O
=	O
1	O
,	O
then	O
the	O
sum	O
of	O
the	O
series	O
can	O
be	O
defined	O
to	O
be	O
the	O
value	O
at	O
x	O
=	O
1	O
.	O
</s>
<s>
Euler	O
summation	O
is	O
essentially	O
an	O
explicit	O
form	O
of	O
analytic	B-Algorithm
continuation	I-Algorithm
.	O
</s>
<s>
If	O
a	O
power	O
series	O
converges	O
for	O
small	O
complex	O
z	O
and	O
can	O
be	O
analytically	B-Algorithm
continued	I-Algorithm
to	O
the	O
open	O
disk	O
with	O
diameter	O
from	O
to	O
1	O
and	O
is	O
continuous	O
at	O
1	O
,	O
then	O
its	O
value	O
at	O
q	O
is	O
called	O
the	O
Euler	O
or	O
(	O
E	O
,	O
q	O
)	O
sum	O
of	O
the	O
series	O
Σan	O
.	O
</s>
<s>
Euler	O
used	O
it	O
before	O
analytic	B-Algorithm
continuation	I-Algorithm
was	O
defined	O
in	O
general	O
,	O
and	O
gave	O
explicit	O
formulas	O
for	O
the	O
power	O
series	O
of	O
the	O
analytic	B-Algorithm
continuation	I-Algorithm
.	O
</s>
<s>
The	O
operation	O
of	O
Euler	O
summation	O
can	O
be	O
repeated	O
several	O
times	O
,	O
and	O
this	O
is	O
essentially	O
equivalent	O
to	O
taking	O
an	O
analytic	B-Algorithm
continuation	I-Algorithm
of	O
a	O
power	O
series	O
to	O
the	O
pointz	O
=	O
1	O
.	O
</s>
<s>
(	O
for	O
positive	O
values	O
of	O
the	O
an	O
)	O
converges	O
for	O
large	O
real	O
s	O
and	O
can	O
be	O
analytically	B-Algorithm
continued	I-Algorithm
along	O
the	O
real	O
line	O
to	O
s=	O
−1	O
,	O
then	O
its	O
value	O
at	O
s=	O
−1	O
is	O
called	O
the	O
zeta	O
regularized	O
sum	O
of	O
the	O
series	O
a1+a2+	O
...	O
Zeta	O
function	O
regularization	O
is	O
nonlinear	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
A	O
has	O
eigenvalues	O
1	O
,	O
2	O
,	O
3	O
,	O
...	O
then	O
f(s )	O
is	O
the	O
Riemann	O
zeta	O
function	O
,	O
ζ(s )	O
,	O
whose	O
value	O
at	O
s=	O
−1	O
is	O
−	O
,	O
assigning	O
a	O
value	O
to	O
the	O
divergent	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
if	O
this	O
limit	B-Algorithm
exists	O
.	O
</s>
<s>
There	O
is	O
a	O
variation	O
of	O
this	O
method	O
where	O
the	O
series	O
for	O
J	O
has	O
a	O
finite	O
radius	O
of	O
convergence	B-Algorithm
r	O
and	O
diverges	O
at	O
x	O
=	O
r	O
.	O
</s>
<s>
In	O
this	O
case	O
one	O
defines	O
the	O
sum	O
as	O
above	O
,	O
except	O
taking	O
the	O
limit	B-Algorithm
as	O
x	O
tends	O
to	O
r	O
rather	O
than	O
infinity	O
.	O
</s>
<s>
A	O
further	O
generalization	O
is	O
to	O
replace	O
the	O
sum	O
under	O
the	O
integral	O
by	O
its	O
analytic	B-Algorithm
continuation	I-Algorithm
from	O
smallt	O
.	O
</s>
<s>
This	O
summation	B-Algorithm
method	I-Algorithm
works	O
by	O
using	O
an	O
extension	O
to	O
the	O
real	O
numbers	O
known	O
as	O
the	O
hyperreal	O
numbers	O
.	O
</s>
<s>
Since	O
the	O
hyperreal	O
numbers	O
include	O
distinct	O
infinite	O
values	O
,	O
these	O
numbers	O
can	O
be	O
used	O
to	O
represent	O
the	O
values	O
of	O
divergent	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
In	O
1812	O
Hutton	O
introduced	O
a	O
method	O
of	O
summing	O
divergent	B-Algorithm
series	I-Algorithm
by	O
starting	O
with	O
the	O
sequence	O
of	O
partial	O
sums	O
,	O
and	O
repeatedly	O
applying	O
the	O
operation	O
of	O
replacing	O
a	O
sequences0	O
,	O
s1	O
,...	O
by	O
the	O
sequence	O
of	O
averages	O
,	O
,...,	O
and	O
then	O
taking	O
the	O
limit	B-Algorithm
.	O
</s>
<s>
Albert	O
Ingham	O
showed	O
that	O
if	O
δ	O
is	O
any	O
positive	O
number	O
then	O
(	O
C	O
,	O
−δ	O
)	O
(	O
Cesàro	B-Algorithm
)	O
summability	B-Algorithm
implies	O
Ingham	O
summability	B-Algorithm
,	O
and	O
Ingham	O
summability	B-Algorithm
implies	O
(	O
C	O
,	O
δ	O
)	O
summability	B-Algorithm
.	O
</s>
<s>
If	O
a	O
series	O
is	O
(	O
C	O
,	O
k	O
)	O
(	O
Cesàro	B-Algorithm
)	O
summable	O
for	O
any	O
k	O
then	O
it	O
is	O
Lambert	O
summable	O
to	O
the	O
same	O
value	O
,	O
and	O
if	O
a	O
series	O
is	O
Lambert	O
summable	O
then	O
it	O
is	O
Abel	O
summable	O
to	O
the	O
same	O
value	O
.	O
</s>
<s>
Ramanujan	O
summation	O
is	O
a	O
method	O
of	O
assigning	O
a	O
value	O
to	O
divergent	B-Algorithm
series	I-Algorithm
used	O
by	O
Ramanujan	O
and	O
based	O
on	O
the	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
summation	I-Algorithm
formula	I-Algorithm
.	O
</s>
<s>
The	O
Ramanujan	O
sum	O
of	O
a	O
series	O
f(0 )	O
+	O
f(1 )	O
+	O
...	O
depends	O
not	O
only	O
on	O
the	O
values	O
of	O
f	O
at	O
integers	O
,	O
but	O
also	O
on	O
values	O
of	O
the	O
function	O
f	O
at	O
non-integral	O
points	O
,	O
so	O
it	O
is	O
not	O
really	O
a	O
summation	B-Algorithm
method	I-Algorithm
in	O
the	O
sense	O
of	O
this	O
article	O
.	O
</s>
