<s>
Ramanujan	B-Algorithm
summation	I-Algorithm
is	O
a	O
technique	O
invented	O
by	O
the	O
mathematician	O
Srinivasa	O
Ramanujan	O
for	O
assigning	O
a	O
value	O
to	O
divergent	B-Algorithm
infinite	O
series	O
.	O
</s>
<s>
Although	O
the	O
Ramanujan	B-Algorithm
summation	I-Algorithm
of	O
a	O
divergent	B-Algorithm
series	I-Algorithm
is	O
not	O
a	O
sum	O
in	O
the	O
traditional	O
sense	O
,	O
it	O
has	O
properties	O
that	O
make	O
it	O
mathematically	O
useful	O
in	O
the	O
study	O
of	O
divergent	B-Algorithm
infinite	O
series	O
,	O
for	O
which	O
conventional	O
summation	O
is	O
undefined	O
.	O
</s>
<s>
Since	O
there	O
are	O
no	O
properties	O
of	O
an	O
entire	O
sum	O
,	O
the	O
Ramanujan	B-Algorithm
summation	I-Algorithm
functions	O
as	O
a	O
property	O
of	O
partial	O
sums	O
.	O
</s>
<s>
If	O
we	O
take	O
the	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
summation	I-Algorithm
formula	I-Algorithm
together	O
with	O
the	O
correction	O
rule	O
using	O
Bernoulli	O
numbers	O
,	O
we	O
see	O
that	O
:	O
</s>
<s>
Comparing	O
both	O
formulae	O
and	O
assuming	O
that	O
R	O
tends	O
to	O
0	O
as	O
x	O
tends	O
to	O
infinity	O
,	O
we	O
see	O
that	O
,	O
in	O
a	O
general	O
case	O
,	O
for	O
functions	O
f(x )	O
with	O
no	O
divergence	B-Algorithm
at	O
x	O
=	O
0	O
:	O
</s>
<s>
For	O
functions	O
f(x )	O
with	O
no	O
divergence	B-Algorithm
at	O
x	O
=	O
1	O
,	O
we	O
obtain	O
:	O
</s>
<s>
C(0 )	O
was	O
then	O
proposed	O
to	O
use	O
as	O
the	O
sum	O
of	O
the	O
divergent	B-Algorithm
sequence	O
.	O
</s>
<s>
In	O
the	O
following	O
text	O
,	O
indicates	O
"	O
Ramanujan	B-Algorithm
summation	I-Algorithm
"	O
.	O
</s>
<s>
Ramanujan	O
had	O
calculated	O
"	O
sums	O
"	O
of	O
known	O
divergent	B-Algorithm
series	I-Algorithm
.	O
</s>
