<s>
The	O
infinite	O
series	O
whose	O
terms	O
are	O
the	O
natural	O
numbers	O
is	O
a	O
divergent	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
which	O
increases	O
without	O
bound	O
as	O
n	O
goes	O
to	O
infinity	B-Application
.	O
</s>
<s>
Because	O
the	O
sequence	O
of	O
partial	O
sums	O
fails	O
to	O
converge	B-Algorithm
to	I-Algorithm
a	I-Algorithm
finite	I-Algorithm
limit	I-Algorithm
,	O
the	O
series	O
does	O
not	O
have	O
a	O
sum	O
.	O
</s>
<s>
For	O
example	O
,	O
many	O
summation	B-Algorithm
methods	I-Algorithm
are	O
used	O
in	O
mathematics	O
to	O
assign	O
numerical	O
values	O
even	O
to	O
a	O
divergent	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
where	O
the	O
left-hand	O
side	O
has	O
to	O
be	O
interpreted	O
as	O
being	O
the	O
value	O
obtained	O
by	O
using	O
one	O
of	O
the	O
aforementioned	O
summation	B-Algorithm
methods	I-Algorithm
and	O
not	O
as	O
the	O
sum	O
of	O
an	O
infinite	O
series	O
in	O
its	O
usual	O
meaning	O
.	O
</s>
<s>
The	O
divergence	B-Algorithm
is	O
a	O
simple	O
consequence	O
of	O
the	O
form	O
of	O
the	O
series	O
:	O
the	O
terms	O
do	O
not	O
approach	O
zero	O
,	O
so	O
the	O
series	O
diverges	O
by	O
the	O
term	O
test	O
.	O
</s>
<s>
Among	O
the	O
classical	O
divergent	B-Algorithm
series	I-Algorithm
,	O
is	O
relatively	O
difficult	O
to	O
manipulate	O
into	O
a	O
finite	O
value	O
.	O
</s>
<s>
Many	O
summation	B-Algorithm
methods	I-Algorithm
are	O
used	O
to	O
assign	O
numerical	O
values	O
to	O
divergent	B-Algorithm
series	I-Algorithm
,	O
some	O
more	O
powerful	O
than	O
others	O
.	O
</s>
<s>
For	O
example	O
,	O
Cesàro	B-Algorithm
summation	I-Algorithm
is	O
a	O
well-known	O
method	O
that	O
sums	O
Grandi	O
's	O
series	O
,	O
the	O
mildly	O
divergent	B-Algorithm
series	I-Algorithm
,	O
to	O
.	O
</s>
<s>
Unlike	O
the	O
above	O
series	O
,	O
is	O
not	O
Cesàro	B-Algorithm
summable	O
nor	O
Abel	O
summable	O
.	O
</s>
<s>
Those	O
methods	O
work	O
on	O
oscillating	O
divergent	B-Algorithm
series	I-Algorithm
,	O
but	O
they	O
cannot	O
produce	O
a	O
finite	O
answer	O
for	O
a	O
series	O
that	O
diverges	O
to	O
+∞	O
.	O
</s>
<s>
Most	O
of	O
the	O
more	O
elementary	O
definitions	O
of	O
the	O
sum	O
of	O
a	O
divergent	B-Algorithm
series	I-Algorithm
are	O
stable	O
and	O
linear	O
,	O
and	O
any	O
method	O
that	O
is	O
both	O
stable	O
and	O
linear	O
cannot	O
sum	O
to	O
a	O
finite	O
value	O
;	O
see	O
below	O
.	O
</s>
<s>
More	O
advanced	O
methods	O
are	O
required	O
,	O
such	O
as	O
zeta	O
function	O
regularization	O
or	O
Ramanujan	B-Algorithm
summation	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
if	O
zeroes	O
are	O
inserted	O
into	O
arbitrary	O
positions	O
of	O
a	O
divergent	B-Algorithm
series	I-Algorithm
,	O
it	O
is	O
possible	O
to	O
arrive	O
at	O
results	O
that	O
are	O
not	O
self-consistent	O
,	O
let	O
alone	O
consistent	O
with	O
other	O
methods	O
.	O
</s>
<s>
The	O
benefit	O
of	O
introducing	O
the	O
Riemann	O
zeta	O
function	O
is	O
that	O
it	O
can	O
be	O
defined	O
for	O
other	O
values	O
of	O
s	O
by	O
analytic	B-Algorithm
continuation	I-Algorithm
.	O
</s>
<s>
The	O
identity	O
continues	O
to	O
hold	O
when	O
both	O
functions	O
are	O
extended	O
by	O
analytic	B-Algorithm
continuation	I-Algorithm
to	O
include	O
values	O
of	O
s	O
for	O
which	O
the	O
above	O
series	O
diverge	O
.	O
</s>
<s>
Substituting	O
,	O
one	O
gets	O
Now	O
,	O
computing	O
η( −1	O
)	O
is	O
an	O
easier	O
task	O
,	O
as	O
the	O
eta	O
function	O
is	O
equal	O
to	O
the	O
Abel	O
sum	O
of	O
its	O
defining	O
series	O
,	O
which	O
is	O
a	O
one-sided	B-Algorithm
limit	I-Algorithm
:	O
</s>
<s>
Smoothing	O
is	O
a	O
conceptual	O
bridge	O
between	O
zeta	O
function	O
regularization	O
,	O
with	O
its	O
reliance	O
on	O
complex	O
analysis	O
,	O
and	O
Ramanujan	B-Algorithm
summation	I-Algorithm
,	O
with	O
its	O
shortcut	O
to	O
the	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
formula	I-Algorithm
.	O
</s>
<s>
One	O
can	O
then	O
prove	O
that	O
this	O
smoothed	O
sum	O
is	O
asymptotic	O
to	O
,	O
where	O
C	O
is	O
a	O
constant	O
that	O
depends	O
on	O
f	O
.	O
The	O
constant	O
term	O
of	O
the	O
asymptotic	O
expansion	O
does	O
not	O
depend	O
on	O
f	O
:	O
it	O
is	O
necessarily	O
the	O
same	O
value	O
given	O
by	O
analytic	B-Algorithm
continuation	I-Algorithm
,	O
.	O
</s>
<s>
The	O
Ramanujan	B-Algorithm
sum	I-Algorithm
of	O
is	O
also	O
.	O
</s>
<s>
I	O
was	O
expecting	O
a	O
reply	O
from	O
you	O
similar	O
to	O
the	O
one	O
which	O
a	O
Mathematics	O
Professor	O
at	O
London	O
wrote	O
asking	O
me	O
to	O
study	O
carefully	O
Bromwich	O
's	O
Infinite	O
Series	O
and	O
not	O
fall	O
into	O
the	O
pitfalls	O
of	O
divergent	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
Ramanujan	B-Algorithm
summation	I-Algorithm
is	O
a	O
method	O
to	O
isolate	O
the	O
constant	O
term	O
in	O
the	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
formula	I-Algorithm
for	O
the	O
partial	O
sums	O
of	O
a	O
series	O
.	O
</s>
<s>
To	O
avoid	O
inconsistencies	O
,	O
the	O
modern	O
theory	O
of	O
Ramanujan	B-Algorithm
summation	I-Algorithm
requires	O
that	O
f	O
is	O
"	O
regular	O
"	O
in	O
the	O
sense	O
that	O
the	O
higher-order	O
derivatives	O
of	O
f	O
decay	O
quickly	O
enough	O
for	O
the	O
remainder	O
terms	O
in	O
the	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
formula	I-Algorithm
to	O
tend	O
to	O
0	O
.	O
</s>
<s>
The	O
regularity	O
requirement	O
prevents	O
the	O
use	O
of	O
Ramanujan	B-Algorithm
summation	I-Algorithm
upon	O
spaced-out	O
series	O
like	O
,	O
because	O
no	O
regular	O
function	O
takes	O
those	O
values	O
.	O
</s>
<s>
A	O
summation	B-Algorithm
method	I-Algorithm
that	O
is	O
linear	O
and	O
stable	O
cannot	O
sum	O
the	O
series	O
to	O
any	O
finite	O
value	O
.	O
</s>
<s>
So	O
using	O
the	O
divergent	B-Algorithm
series	I-Algorithm
,	O
the	O
sum	O
over	O
all	O
harmonics	O
is	O
.	O
</s>
<s>
According	O
to	O
Morris	O
Kline	O
,	O
Euler	O
's	O
early	O
work	O
on	O
divergent	B-Algorithm
series	I-Algorithm
relied	O
on	O
function	O
expansions	O
,	O
from	O
which	O
he	O
concluded	O
.	O
</s>
<s>
The	O
main	O
character	O
,	O
Ruth	O
,	O
walks	O
into	O
a	O
lecture	O
hall	O
and	O
introduces	O
the	O
idea	O
of	O
a	O
divergent	B-Algorithm
series	I-Algorithm
before	O
proclaiming	O
,	O
"	O
I	O
'm	O
going	O
to	O
show	O
you	O
something	O
really	O
thrilling	O
"	O
,	O
namely	O
.	O
</s>
<s>
In	O
January	O
2014	O
,	O
Numberphile	O
produced	O
a	O
YouTube	B-General_Concept
video	I-General_Concept
on	O
the	O
series	O
,	O
which	O
gathered	O
over	O
1.5million	O
views	O
in	O
its	O
first	O
month	O
.	O
</s>
<s>
After	O
receiving	O
complaints	O
about	O
the	O
lack	O
of	O
rigour	O
in	O
the	O
first	O
video	O
,	O
Padilla	O
also	O
wrote	O
an	O
explanation	O
on	O
his	O
webpage	O
relating	O
the	O
manipulations	O
in	O
the	O
video	O
to	O
identities	O
between	O
the	O
analytic	B-Algorithm
continuations	I-Algorithm
of	O
the	O
relevant	O
Dirichlet	O
series	O
.	O
</s>
<s>
Coverage	O
of	O
this	O
topic	O
in	O
Smithsonian	O
magazine	O
describes	O
the	O
Numberphile	O
video	O
as	O
misleading	O
and	O
notes	O
that	O
the	O
interpretation	O
of	O
the	O
sum	O
as	O
relies	O
on	O
a	O
specialized	O
meaning	O
for	O
the	O
equals	O
sign	O
,	O
from	O
the	O
techniques	O
of	O
analytic	B-Algorithm
continuation	I-Algorithm
,	O
in	O
which	O
equals	O
means	O
is	O
associated	O
with	O
.	O
</s>
