<s>
A	O
series	O
is	O
convergent	O
(	O
or	O
converges	O
)	O
if	O
the	O
sequence	O
of	O
its	O
partial	O
sums	O
tends	O
to	O
a	O
limit	B-Algorithm
;	O
that	O
means	O
that	O
,	O
when	O
adding	O
one	O
after	O
the	O
other	O
in	O
the	O
order	O
given	O
by	O
the	O
indices	O
,	O
one	O
gets	O
partial	O
sums	O
that	O
become	O
closer	O
and	O
closer	O
to	O
a	O
given	O
number	O
.	O
</s>
<s>
Any	O
series	O
that	O
is	O
not	O
convergent	O
is	O
said	O
to	O
be	O
divergent	B-Algorithm
or	O
to	O
diverge	O
.	O
</s>
<s>
The	O
reciprocals	O
of	O
the	O
positive	O
integers	O
produce	O
a	O
divergent	B-Algorithm
series	I-Algorithm
(	O
harmonic	B-Algorithm
series	I-Algorithm
)	O
:	O
</s>
<s>
Alternating	O
the	O
signs	O
of	O
the	O
reciprocals	O
of	O
positive	O
integers	O
produces	O
a	O
convergent	B-Algorithm
series	I-Algorithm
(	O
alternating	O
harmonic	B-Algorithm
series	I-Algorithm
)	O
:	O
</s>
<s>
The	O
reciprocals	O
of	O
prime	O
numbers	O
produce	O
a	O
divergent	B-Algorithm
series	I-Algorithm
(	O
so	O
the	O
set	O
of	O
primes	O
is	O
"	O
large	O
"	O
;	O
see	O
divergence	B-Algorithm
of	O
the	O
sum	O
of	O
the	O
reciprocals	O
of	O
the	O
primes	O
)	O
:	O
</s>
<s>
The	O
reciprocals	O
of	O
triangular	O
numbers	O
produce	O
a	O
convergent	B-Algorithm
series	I-Algorithm
:	O
</s>
<s>
The	O
reciprocals	O
of	O
factorials	O
produce	O
a	O
convergent	B-Algorithm
series	I-Algorithm
(	O
see	O
Euler	O
's	O
number	O
)	O
:	O
</s>
<s>
The	O
reciprocals	O
of	O
square	O
numbers	O
produce	O
a	O
convergent	B-Algorithm
series	I-Algorithm
(	O
the	O
Basel	O
problem	O
)	O
:	O
</s>
<s>
The	O
reciprocals	O
of	O
powers	O
of	O
2	O
produce	O
a	O
convergent	B-Algorithm
series	I-Algorithm
(	O
so	O
the	O
set	O
of	O
powers	O
of	O
2	O
is	O
"	O
small	O
"	O
)	O
:	O
</s>
<s>
The	O
reciprocals	O
of	O
powers	O
of	O
any	O
n>1	O
produce	O
a	O
convergent	B-Algorithm
series	I-Algorithm
:	O
</s>
<s>
Alternating	O
the	O
signs	O
of	O
reciprocals	O
of	O
powers	O
of	O
2	O
also	O
produces	O
a	O
convergent	B-Algorithm
series	I-Algorithm
:	O
</s>
<s>
Alternating	O
the	O
signs	O
of	O
reciprocals	O
of	O
powers	O
of	O
any	O
n>1	O
produces	O
a	O
convergent	B-Algorithm
series	I-Algorithm
:	O
</s>
<s>
The	O
reciprocals	O
of	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
produce	O
a	O
convergent	B-Algorithm
series	I-Algorithm
(	O
see	O
ψ	O
)	O
:	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
methods	O
of	O
determining	O
whether	O
a	O
series	O
converges	O
or	O
diverges	B-Algorithm
.	O
</s>
<s>
If	O
then	O
the	O
series	O
diverges	B-Algorithm
.	O
</s>
<s>
where	O
"	O
lim	O
sup	O
"	O
denotes	O
the	O
limit	B-Algorithm
superior	O
(	O
possibly	O
∞	O
;	O
if	O
the	O
limit	B-Algorithm
exists	O
it	O
is	O
the	O
same	O
value	O
)	O
.	O
</s>
<s>
If	O
then	O
the	O
series	O
diverges	B-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
if	O
the	O
ratio	O
test	O
works	O
(	O
meaning	O
that	O
the	O
limit	B-Algorithm
exists	O
and	O
is	O
not	O
equal	O
to	O
1	O
)	O
then	O
so	O
does	O
the	O
root	O
test	O
;	O
the	O
converse	O
,	O
however	O
,	O
is	O
not	O
true	O
.	O
</s>
<s>
The	O
root	O
test	O
is	O
therefore	O
more	O
generally	O
applicable	O
,	O
but	O
as	O
a	O
practical	O
matter	O
the	O
limit	B-Algorithm
is	O
often	O
difficult	O
to	O
compute	O
for	O
commonly	O
seen	O
types	O
of	O
series	O
.	O
</s>
<s>
The	O
series	O
can	O
be	O
compared	O
to	O
an	O
integral	O
to	O
establish	O
convergence	B-Algorithm
or	O
divergence	B-Algorithm
.	O
</s>
<s>
But	O
if	O
the	O
integral	O
diverges	B-Algorithm
,	O
then	O
the	O
series	O
does	O
so	O
as	O
well	O
.	O
</s>
<s>
Limit	B-Algorithm
comparison	O
test	O
.	O
</s>
<s>
If	O
,	O
and	O
the	O
limit	B-Algorithm
exists	O
and	O
is	O
not	O
zero	O
,	O
then	O
converges	O
if	O
and	O
only	O
if	O
converges	O
.	O
</s>
<s>
Also	O
known	O
as	O
the	O
Leibniz	O
criterion	O
,	O
the	O
alternating	O
series	O
test	O
states	O
that	O
for	O
an	O
alternating	O
series	O
of	O
the	O
form	O
,	O
if	O
is	O
monotonically	O
decreasing	O
,	O
and	O
has	O
a	O
limit	B-Algorithm
of	O
0	O
at	O
infinity	O
,	O
then	O
the	O
series	O
converges	O
.	O
</s>
<s>
If	O
the	O
series	O
converges	O
but	O
the	O
series	O
diverges	B-Algorithm
,	O
then	O
the	O
series	O
is	O
conditionally	O
convergent	O
.	O
</s>
<s>
The	O
Riemann	O
series	O
theorem	O
states	O
that	O
if	O
a	O
series	O
converges	O
conditionally	O
,	O
it	O
is	O
possible	O
to	O
rearrange	O
the	O
terms	O
of	O
the	O
series	O
in	O
such	O
a	O
way	O
that	O
the	O
series	O
converges	O
to	O
any	O
value	O
,	O
or	O
even	O
diverges	B-Algorithm
.	O
</s>
<s>
There	O
is	O
an	O
analogue	O
of	O
the	O
comparison	O
test	O
for	O
infinite	O
series	O
of	O
functions	O
called	O
the	B-Algorithm
Weierstrass	I-Algorithm
M-test	I-Algorithm
.	O
</s>
