<s>
In	O
mathematical	O
analysis	O
,	O
Cesàro	B-Algorithm
summation	I-Algorithm
(	O
also	O
known	O
as	O
the	O
Cesàro	B-Algorithm
mean	I-Algorithm
)	O
assigns	O
values	O
to	O
some	O
infinite	O
sums	O
that	O
are	O
not	B-Algorithm
necessarily	I-Algorithm
convergent	I-Algorithm
in	O
the	O
usual	O
sense	O
.	O
</s>
<s>
The	O
Cesàro	B-Algorithm
sum	I-Algorithm
is	O
defined	O
as	O
the	O
limit	O
,	O
as	O
n	O
tends	O
to	O
infinity	O
,	O
of	O
the	O
sequence	O
of	O
arithmetic	O
means	O
of	O
the	O
first	O
n	O
partial	O
sums	O
of	O
the	O
series	O
.	O
</s>
<s>
This	O
special	O
case	O
of	O
a	O
matrix	O
summability	B-Algorithm
method	I-Algorithm
is	O
named	O
for	O
the	O
Italian	O
analyst	O
Ernesto	O
Cesàro	B-Algorithm
(	O
1859	O
–	O
1906	O
)	O
.	O
</s>
<s>
The	O
term	O
summation	O
can	O
be	O
misleading	O
,	O
as	O
some	O
statements	O
and	O
proofs	O
regarding	O
Cesàro	B-Algorithm
summation	I-Algorithm
can	O
be	O
said	O
to	O
implicate	O
the	O
Eilenberg	O
–	O
Mazur	O
swindle	O
.	O
</s>
<s>
The	O
sequence	O
is	O
called	O
Cesàro	B-Algorithm
summable	O
,	O
with	O
Cesàro	B-Algorithm
sum	I-Algorithm
,	O
if	O
,	O
as	O
tends	O
to	O
infinity	O
,	O
the	O
arithmetic	O
mean	O
of	O
its	O
first	O
n	O
partial	O
sums	O
tends	O
to	O
:	O
</s>
<s>
The	O
value	O
of	O
the	O
resulting	O
limit	O
is	O
called	O
the	O
Cesàro	B-Algorithm
sum	I-Algorithm
of	O
the	O
series	O
If	O
this	O
series	O
is	O
convergent	O
,	O
then	O
it	O
is	O
Cesàro	B-Algorithm
summable	O
and	O
its	O
Cesàro	B-Algorithm
sum	I-Algorithm
is	O
the	O
usual	O
sum	O
.	O
</s>
<s>
However	O
,	O
Cesàro	B-Algorithm
summable	O
.	O
</s>
<s>
and	O
therefore	O
,	O
the	O
Cesàro	B-Algorithm
sum	I-Algorithm
of	O
the	O
series	O
is	O
.	O
</s>
<s>
This	O
sequence	O
diverges	O
to	O
infinity	O
as	O
well	O
,	O
so	O
is	O
Cesàro	B-Algorithm
summable	O
.	O
</s>
<s>
In	O
fact	O
,	O
for	O
any	O
sequence	O
which	O
diverges	O
to	O
(	O
positive	O
or	O
negative	O
)	O
infinity	O
,	O
the	O
Cesàro	B-Algorithm
method	O
also	O
leads	O
to	O
a	O
sequence	O
that	O
diverges	O
likewise	O
,	O
and	O
hence	O
such	O
a	O
series	O
is	O
not	O
Cesàro	B-Algorithm
summable	O
.	O
</s>
<s>
In	O
1890	O
,	O
Ernesto	O
Cesàro	B-Algorithm
stated	O
a	O
broader	O
family	O
of	O
summation	B-Algorithm
methods	I-Algorithm
which	O
have	O
since	O
been	O
called	O
for	O
non-negative	O
integers	O
.	O
</s>
<s>
The	O
method	O
is	O
just	O
ordinary	O
summation	O
,	O
and	O
is	O
Cesàro	B-Algorithm
summation	I-Algorithm
as	O
described	O
above	O
.	O
</s>
