<s>
Two	O
of	O
the	O
simplest	O
methods	O
are	O
Cesàro	B-Algorithm
summation	I-Algorithm
and	O
Abel	O
summation	O
.	O
</s>
<s>
The	O
first	O
rigorous	O
method	O
for	O
summing	O
divergent	O
series	O
was	O
published	O
by	O
Ernesto	O
Cesàro	B-Algorithm
in	O
1890	O
.	O
</s>
<s>
The	O
basic	O
idea	O
is	O
similar	O
to	O
Leibniz	O
's	O
probabilistic	O
approach	O
:	O
essentially	O
,	O
the	O
Cesàro	B-Algorithm
sum	I-Algorithm
of	O
a	O
series	O
is	O
the	O
average	O
of	O
all	O
of	O
its	O
partial	O
sums	O
.	O
</s>
<s>
Formally	O
one	O
computes	O
,	O
for	O
each	O
n	O
,	O
the	O
average	O
σn	O
of	O
the	O
first	O
n	O
partial	O
sums	O
,	O
and	O
takes	O
the	O
limit	O
of	O
these	O
Cesàro	B-Algorithm
means	I-Algorithm
as	O
n	O
goes	O
to	O
infinity	O
.	O
</s>
<s>
This	O
sequence	O
of	O
arithmetic	O
means	O
converges	O
to	O
1⁄2	O
,	O
so	O
the	O
Cesàro	B-Algorithm
sum	I-Algorithm
of	O
Σak	O
is	O
1⁄2	O
.	O
</s>
<s>
Equivalently	O
,	O
one	O
says	O
that	O
the	O
Cesàro	B-Algorithm
limit	O
of	O
the	O
sequence	O
1	O
,	O
0	O
,	O
1	O
,	O
0	O
,	O
…	O
is	O
1⁄2	O
.	O
</s>
<s>
The	O
Cesàro	B-Algorithm
sum	I-Algorithm
of	O
1	O
+	O
0	O
−	O
1	O
+	O
1	O
+	O
0	O
−	O
1	O
+	O
·	O
·	O
·	O
is	O
2⁄3	O
.	O
</s>
<s>
So	O
the	O
Cesàro	B-Algorithm
sum	I-Algorithm
of	O
a	O
series	O
can	O
be	O
altered	O
by	O
inserting	O
infinitely	O
many	O
0s	O
as	O
well	O
as	O
infinitely	O
many	O
brackets	O
.	O
</s>
<s>
Whenever	O
a	O
series	O
is	O
Cesàro	B-Algorithm
summable	O
,	O
it	O
is	O
also	O
Abel	O
summable	O
and	O
has	O
the	O
same	O
sum	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
taking	O
the	O
Cauchy	O
product	O
of	O
Grandi	O
's	O
series	O
with	O
itself	O
yields	O
a	O
series	O
which	O
is	O
Abel	O
summable	O
but	O
not	O
Cesàro	B-Algorithm
summable	O
:	O
</s>
<s>
If	O
φ	O
is	O
additionally	O
assumed	O
to	O
be	O
continuously	O
differentiable	O
,	O
then	O
the	O
claim	O
can	O
be	O
proved	O
by	O
applying	O
the	O
mean	B-Algorithm
value	I-Algorithm
theorem	I-Algorithm
and	O
converting	O
the	O
sum	O
into	O
an	O
integral	O
.	O
</s>
