<s>
In	O
mathematics	O
,	O
uniform	B-Algorithm
absolute-convergence	I-Algorithm
is	O
a	O
type	O
of	O
convergence	B-Algorithm
for	O
series	O
of	O
functions	O
.	O
</s>
<s>
Like	O
absolute-convergence	O
,	O
it	O
has	O
the	O
useful	O
property	O
that	O
it	O
is	O
preserved	O
when	O
the	O
order	O
of	O
summation	O
is	O
changed	O
.	O
</s>
<s>
A	O
convergent	B-Algorithm
series	I-Algorithm
of	O
numbers	O
can	O
often	O
be	O
reordered	O
in	O
such	O
a	O
way	O
that	O
the	O
new	O
series	O
diverges	O
.	O
</s>
<s>
This	O
is	O
not	O
possible	O
for	O
series	O
of	O
nonnegative	O
numbers	O
,	O
however	O
,	O
so	O
the	O
notion	O
of	O
absolute-convergence	O
precludes	O
this	O
phenomenon	O
.	O
</s>
<s>
When	O
dealing	O
with	O
uniformly	B-Algorithm
convergent	I-Algorithm
series	O
of	O
functions	O
,	O
the	O
same	O
phenomenon	O
occurs	O
:	O
the	O
series	O
can	O
potentially	O
be	O
reordered	O
into	O
a	O
non-uniformly	O
convergent	O
series	O
,	O
or	O
a	O
series	O
which	O
does	O
not	O
even	O
converge	O
pointwise	O
.	O
</s>
<s>
This	O
is	O
impossible	O
for	O
series	O
of	O
nonnegative	O
functions	O
,	O
so	O
the	O
notion	O
of	O
uniform	B-Algorithm
absolute-convergence	I-Algorithm
can	O
be	O
used	O
to	O
rule	O
out	O
these	O
possibilities	O
.	O
</s>
<s>
is	O
uniformly	B-Algorithm
convergent	I-Algorithm
.	O
</s>
<s>
A	O
series	O
can	O
be	O
uniformly	B-Algorithm
convergent	I-Algorithm
and	O
absolutely	O
convergent	O
without	O
being	O
uniformly	O
absolutely-convergent	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
ƒn(x )	O
=	O
xn/n	O
on	O
the	O
open	O
interval	O
(	O
1	O
,	O
0	O
)	O
,	O
then	O
the	O
series	O
Σfn(x )	O
converges	B-Algorithm
uniformly	I-Algorithm
by	O
comparison	O
of	O
the	O
partial	O
sums	O
to	O
those	O
of	O
Σ(1 )	O
n/n	O
,	O
and	O
the	O
series	O
Σ|	O
fn(x )	O
|	O
converges	O
absolutely	O
at	O
each	O
point	O
by	O
the	O
geometric	O
series	O
test	O
,	O
but	O
Σ|	O
fn(x )	O
|	O
does	O
not	O
converge	O
uniformly	O
.	O
</s>
<s>
Intuitively	O
,	O
this	O
is	O
because	O
the	O
absolute-convergence	O
gets	O
slower	O
and	O
slower	O
as	O
x	O
approaches	O
1	O
,	O
where	O
convergence	B-Algorithm
holds	O
but	O
absolute	O
convergence	B-Algorithm
fails	O
.	O
</s>
<s>
If	O
a	O
series	O
of	O
functions	O
into	O
C	O
(	O
or	O
any	O
Banach	O
space	O
)	O
is	O
uniformly	O
absolutely-convergent	O
,	O
then	O
it	O
is	O
uniformly	B-Algorithm
convergent	I-Algorithm
.	O
</s>
<s>
Uniform	B-Algorithm
absolute-convergence	I-Algorithm
is	O
independent	O
of	O
the	O
ordering	O
of	O
a	O
series	O
.	O
</s>
<s>
This	O
is	O
because	O
,	O
for	O
a	O
series	O
of	O
nonnegative	O
functions	O
,	O
uniform	B-Algorithm
convergence	I-Algorithm
is	O
equivalent	O
to	O
the	O
property	O
that	O
,	O
for	O
any	O
ε	O
0	O
,	O
there	O
are	O
finitely	O
many	O
terms	O
of	O
the	O
series	O
such	O
that	O
excluding	O
these	O
terms	O
results	O
in	O
a	O
series	O
with	O
total	O
sum	O
less	O
than	O
the	O
constant	O
function	O
ε	O
,	O
and	O
this	O
property	O
does	O
not	O
refer	O
to	O
the	O
ordering	O
.	O
</s>
