<s>
In	O
mathematics	O
,	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
is	O
the	O
infinite	O
series	O
formed	O
by	O
summing	O
all	O
positive	O
unit	O
fractions	O
:	O
</s>
<s>
Because	O
the	O
logarithm	O
has	O
arbitrarily	O
large	O
values	O
,	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
does	O
not	O
have	O
a	O
finite	O
limit	O
:	O
it	O
is	O
a	O
divergent	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
Its	O
divergence	B-Algorithm
was	O
proven	O
in	O
the	O
14th	O
century	O
by	O
Nicole	O
Oresme	O
using	O
a	O
precursor	O
to	O
the	O
Cauchy	O
condensation	O
test	O
for	O
the	O
convergence	O
of	O
infinite	O
series	O
.	O
</s>
<s>
Applications	O
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
and	O
its	O
partial	O
sums	O
include	O
Euler	O
's	O
proof	O
that	O
there	O
are	O
infinitely	O
many	O
prime	O
numbers	O
,	O
the	O
analysis	O
of	O
the	O
coupon	O
collector	O
's	O
problem	O
on	O
how	O
many	O
random	O
trials	O
are	O
needed	O
to	O
provide	O
a	O
complete	O
range	O
of	O
responses	O
,	O
the	O
connected	O
components	O
of	O
random	O
graphs	O
,	O
the	O
block-stacking	O
problem	O
on	O
how	O
far	O
over	O
the	O
edge	O
of	O
a	O
table	O
a	O
stack	O
of	O
blocks	O
can	O
be	O
cantilevered	O
,	O
and	O
the	O
average	B-General_Concept
case	I-General_Concept
analysis	I-General_Concept
of	O
the	O
quicksort	B-Algorithm
algorithm	O
.	O
</s>
<s>
The	O
name	O
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
derives	O
from	O
the	O
concept	O
of	O
overtones	O
or	O
harmonics	O
in	O
music	O
:	O
the	O
wavelengths	O
of	O
the	O
overtones	O
of	O
a	O
vibrating	O
string	O
are	O
etc.	O
,	O
of	O
the	O
string	O
's	O
fundamental	O
wavelength	O
.	O
</s>
<s>
Every	O
term	O
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
after	O
the	O
first	O
is	O
the	O
harmonic	O
mean	O
of	O
the	O
neighboring	O
terms	O
,	O
so	O
the	O
terms	O
form	O
a	O
harmonic	O
progression	O
;	O
the	O
phrases	O
harmonic	O
mean	O
and	O
harmonic	O
progression	O
likewise	O
derive	O
from	O
music	O
.	O
</s>
<s>
This	O
was	O
so	O
particularly	O
in	O
the	O
Baroque	B-Application
period	O
,	O
when	O
architects	O
used	O
them	O
to	O
establish	O
the	O
proportions	O
of	O
floor	O
plans	O
,	O
of	O
elevations	O
,	O
and	O
to	O
establish	O
harmonic	O
relationships	O
between	O
both	O
interior	O
and	O
exterior	O
architectural	O
details	O
of	O
churches	O
and	O
palaces	O
.	O
</s>
<s>
The	O
divergence	B-Algorithm
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
was	O
first	O
proven	O
in	O
1350	O
by	O
Nicole	O
Oresme	O
.	O
</s>
<s>
The	O
partial	O
sums	O
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
were	O
named	O
harmonic	O
numbers	O
,	O
and	O
given	O
their	O
usual	O
notation	O
,	O
in	O
1968	O
by	O
Donald	O
Knuth	O
.	O
</s>
<s>
It	O
is	O
a	O
divergent	B-Algorithm
series	I-Algorithm
:	O
as	O
more	O
terms	O
of	O
the	O
series	O
are	O
included	O
in	O
partial	O
sums	O
of	O
the	O
series	O
,	O
the	O
values	O
of	O
these	O
partial	O
sums	O
grow	O
arbitrarily	O
large	O
,	O
beyond	O
any	O
finite	O
limit	O
.	O
</s>
<s>
Because	O
it	O
is	O
a	O
divergent	B-Algorithm
series	I-Algorithm
,	O
it	O
should	O
be	O
interpreted	O
as	O
a	O
formal	O
sum	O
,	O
an	O
abstract	O
mathematical	O
expression	O
combining	O
the	O
unit	O
fractions	O
,	O
rather	O
than	O
as	O
something	O
that	O
can	O
be	O
evaluated	O
to	O
a	O
numeric	O
value	O
.	O
</s>
<s>
There	O
are	O
many	O
different	O
proofs	O
of	O
the	O
divergence	B-Algorithm
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
,	O
surveyed	O
in	O
a	O
2006	O
paper	O
by	O
S	O
.	O
J	O
.	O
Kifowit	O
and	O
T	O
.	O
A	O
.	O
Stamps	O
.	O
</s>
<s>
One	O
way	O
to	O
prove	O
divergence	B-Algorithm
is	O
to	O
compare	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
with	O
another	O
divergent	B-Algorithm
series	I-Algorithm
,	O
where	O
each	O
denominator	O
is	O
replaced	O
with	O
the	O
next-largest	O
power	O
of	O
two	O
:	O
</s>
<s>
Because	O
each	O
term	O
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
is	O
greater	O
than	O
or	O
equal	O
to	O
the	O
corresponding	O
term	O
of	O
the	O
second	O
series	O
(	O
and	O
the	O
terms	O
are	O
all	O
positive	O
)	O
,	O
it	O
follows	O
(	O
by	O
the	O
comparison	O
test	O
)	O
that	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
diverges	O
as	O
well	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
prove	O
that	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
diverges	O
by	O
comparing	O
its	O
sum	O
with	O
an	O
improper	O
integral	O
.	O
</s>
<s>
Each	O
rectangle	O
is	O
1	O
unit	O
wide	O
and	O
units	O
high	O
,	O
so	O
if	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
converged	O
then	O
the	O
total	O
area	O
of	O
the	O
rectangles	O
would	O
be	O
the	O
sum	O
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
This	O
shows	O
that	O
the	O
partial	O
sums	O
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
differ	O
from	O
the	O
integral	O
by	O
an	O
amount	O
that	O
is	O
bounded	O
above	O
and	O
below	O
by	O
the	O
unit	O
area	O
of	O
the	O
first	O
rectangle	O
:	O
</s>
<s>
Generalizing	O
this	O
argument	O
,	O
any	O
infinite	O
sum	O
of	O
values	O
of	O
a	O
monotone	O
decreasing	O
positive	O
function	O
(	O
like	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
)	O
has	O
partial	O
sums	O
that	O
are	O
within	O
a	O
bounded	O
distance	O
of	O
the	O
values	O
of	O
the	O
corresponding	O
integrals	O
.	O
</s>
<s>
Partial	O
sum	O
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
,	O
expressed	O
as	O
a	O
fraction	O
decimal	O
relative	O
size1	O
1	O
2	O
3	O
/2	O
3	O
11	O
/6	O
~	O
4	O
25	O
/12	O
~	O
5	O
137	O
/60	O
~	O
6	O
49	O
/20	O
7	O
363	O
/140	O
~	O
8	O
761	O
/280	O
~	O
9	O
7129	O
/2520	O
~	O
10	O
7381	O
/2520	O
~	O
11	O
83711	O
/27720	O
~	O
12	O
86021	O
/27720	O
~	O
13	O
1145993	O
/360360	O
~	O
14	O
1171733	O
/360360	O
~	O
15	O
1195757	O
/360360	O
~	O
16	O
2436559	O
/720720	O
~	O
17	O
42142223	O
/12252240	O
~	O
18	O
14274301	O
/4084080	O
~	O
19	O
275295799	O
/77597520	O
~	O
20	O
55835135	O
/15519504	O
~	O
</s>
<s>
More	O
precisely	O
,	O
by	O
the	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
formula	I-Algorithm
,	O
</s>
<s>
Many	O
well-known	O
mathematical	O
problems	O
have	O
solutions	O
involving	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
and	O
its	O
partial	O
sums	O
.	O
</s>
<s>
The	O
divergence	B-Algorithm
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
implies	O
that	O
crossings	O
of	O
any	O
length	O
are	O
possible	O
with	O
enough	O
fuel	O
.	O
</s>
<s>
The	O
divergence	B-Algorithm
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
implies	O
that	O
there	O
is	O
no	O
limit	O
on	O
how	O
far	O
beyond	O
the	O
table	O
the	O
block	O
stack	O
can	O
extend	O
.	O
</s>
<s>
In	O
1737	O
,	O
Leonhard	O
Euler	O
observed	O
that	O
,	O
as	O
a	O
formal	O
sum	O
,	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
is	O
equal	O
to	O
an	O
Euler	O
product	O
in	O
which	O
each	O
term	O
comes	O
from	O
a	O
prime	O
number	O
:	O
where	O
denotes	O
the	O
set	O
of	O
prime	O
numbers	O
.	O
</s>
<s>
The	O
left	O
equality	O
comes	O
from	O
applying	O
the	O
distributive	O
law	O
to	O
the	O
product	O
and	O
recognizing	O
the	O
resulting	O
terms	O
as	O
the	O
prime	O
factorizations	O
of	O
the	O
terms	O
in	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
,	O
and	O
the	O
right	O
equality	O
uses	O
the	O
standard	O
formula	O
for	O
a	O
geometric	O
series	O
.	O
</s>
<s>
It	O
follows	O
from	O
these	O
manipulations	O
that	O
the	O
sum	O
of	O
reciprocals	O
of	O
primes	O
,	O
on	O
the	O
right	O
hand	O
of	O
this	O
equality	O
,	O
must	O
diverge	O
,	O
for	O
if	O
it	O
converged	O
these	O
steps	O
could	O
be	O
reversed	O
to	O
show	O
that	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
also	O
converges	O
,	O
which	O
it	O
does	O
not	O
.	O
</s>
<s>
Another	O
problem	O
in	O
number	O
theory	O
closely	O
related	O
to	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
concerns	O
the	O
average	O
number	O
of	O
divisors	O
of	O
the	O
numbers	O
in	O
a	O
range	O
from	O
1	O
to	O
,	O
formalized	O
as	O
the	O
average	O
order	O
of	O
the	O
divisor	O
function	O
,	O
</s>
<s>
The	O
operation	O
of	O
rounding	O
each	O
term	O
in	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
to	O
the	O
next	O
smaller	O
integer	O
multiple	O
of	O
causes	O
this	O
average	O
to	O
differ	O
from	O
the	O
harmonic	O
numbers	O
by	O
a	O
small	O
constant	O
,	O
and	O
Peter	O
Gustav	O
Lejeune	O
Dirichlet	O
showed	O
more	O
precisely	O
that	O
the	O
average	O
number	O
of	O
divisors	O
is	O
(	O
expressed	O
in	O
big	O
O	O
notation	O
)	O
.	O
</s>
<s>
The	O
quicksort	B-Algorithm
algorithm	O
for	O
sorting	O
a	O
set	O
of	O
items	O
can	O
be	O
analyzed	O
using	O
the	O
harmonic	O
numbers	O
.	O
</s>
<s>
In	O
either	O
its	O
average-case	B-General_Concept
complexity	O
(	O
with	O
the	O
assumption	O
that	O
all	O
input	O
permutations	O
are	O
equally	O
likely	O
)	O
or	O
in	O
its	O
expected	O
time	O
analysis	O
of	O
worst-case	B-General_Concept
inputs	O
with	O
a	O
random	O
choice	O
of	O
pivot	O
,	O
all	O
of	O
the	O
items	O
are	O
equally	O
likely	O
to	O
be	O
chosen	O
as	O
the	O
pivot	O
.	O
</s>
<s>
The	O
divergence	B-Algorithm
of	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
corresponds	O
in	O
this	O
application	O
to	O
the	O
fact	O
that	O
,	O
in	O
the	O
comparison	B-Algorithm
model	I-Algorithm
of	I-Algorithm
sorting	I-Algorithm
used	O
for	O
quicksort	B-Algorithm
,	O
it	O
is	O
not	O
possible	O
to	O
sort	O
in	O
linear	O
time	O
.	O
</s>
<s>
is	O
known	O
as	O
the	O
alternating	O
harmonic	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
which	O
for	O
would	O
be	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
It	O
can	O
be	O
extended	O
by	O
analytic	B-Algorithm
continuation	I-Algorithm
to	O
a	O
holomorphic	O
function	O
on	O
all	O
complex	O
numbers	O
where	O
the	O
extended	O
function	O
has	O
a	O
simple	O
pole	O
.	O
</s>
<s>
The	O
depleted	O
harmonic	B-Algorithm
series	I-Algorithm
where	O
all	O
of	O
the	O
terms	O
in	O
which	O
the	O
digit	O
9	O
appears	O
anywhere	O
in	O
the	O
denominator	O
are	O
removed	O
can	O
be	O
shown	O
to	O
converge	O
to	O
the	O
value	O
.	O
</s>
