<s>
In	O
mathematics	O
,	O
the	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
formula	I-Algorithm
is	O
a	O
formula	O
for	O
the	O
difference	O
between	O
an	O
integral	O
and	O
a	O
closely	O
related	O
sum	O
.	O
</s>
<s>
For	O
example	O
,	O
many	O
asymptotic	O
expansions	O
are	O
derived	O
from	O
the	O
formula	O
,	O
and	O
Faulhaber	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
for	O
the	O
sum	O
of	O
powers	O
is	O
an	O
immediate	O
consequence	O
.	O
</s>
<s>
It	O
was	O
later	O
generalized	O
to	O
Darboux	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
.	O
</s>
<s>
The	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
formula	I-Algorithm
provides	O
expressions	O
for	O
the	O
difference	O
between	O
the	O
sum	O
and	O
the	O
integral	O
in	O
terms	O
of	O
the	O
higher	O
derivatives	B-Algorithm
evaluated	O
at	O
the	O
endpoints	O
of	O
the	O
interval	O
,	O
that	O
is	O
to	O
say	O
and	O
.	O
</s>
<s>
where	O
is	O
the	O
th	O
Bernoulli	O
number	O
(	O
with	O
)	O
and	O
is	O
an	O
error	B-Algorithm
term	I-Algorithm
which	O
depends	O
on	O
,	O
,	O
,	O
and	O
and	O
is	O
usually	O
small	O
for	O
suitable	O
values	O
of	O
.	O
</s>
<s>
Therefore	O
the	O
low-order	O
cases	O
of	O
the	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
formula	I-Algorithm
are	O
:	O
</s>
<s>
Euler	O
computed	O
this	O
sum	O
to	O
20	O
decimal	O
places	O
with	O
only	O
a	O
few	O
terms	O
of	O
the	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
formula	I-Algorithm
in	O
1735	O
.	O
</s>
<s>
This	O
may	O
be	O
viewed	O
as	O
an	O
extension	O
of	O
the	O
trapezoid	B-Algorithm
rule	I-Algorithm
by	O
the	O
inclusion	O
of	O
correction	O
terms	O
.	O
</s>
<s>
The	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
formula	I-Algorithm
is	O
also	O
used	O
for	O
detailed	O
error	B-Algorithm
analysis	I-Algorithm
in	O
numerical	B-Algorithm
quadrature	I-Algorithm
.	O
</s>
<s>
It	O
explains	O
the	O
superior	O
performance	O
of	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
on	O
smooth	O
periodic	O
functions	O
and	O
is	O
used	O
in	O
certain	O
extrapolation	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Clenshaw	B-Algorithm
–	I-Algorithm
Curtis	I-Algorithm
quadrature	I-Algorithm
is	O
essentially	O
a	O
change	O
of	O
variables	O
to	O
cast	O
an	O
arbitrary	O
integral	O
in	O
terms	O
of	O
integrals	O
of	O
periodic	O
functions	O
where	O
the	O
Euler	O
–	O
Maclaurin	O
approach	O
is	O
very	O
accurate	O
(	O
in	O
that	O
particular	O
case	O
the	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
formula	I-Algorithm
takes	O
the	O
form	O
of	O
a	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
)	O
.	O
</s>
<s>
That	O
expansion	O
,	O
in	O
turn	O
,	O
serves	O
as	O
the	O
starting	O
point	O
for	O
one	O
of	O
the	O
derivations	B-Algorithm
of	O
precise	O
error	O
estimates	O
for	O
Stirling	O
's	O
approximation	O
of	O
the	O
factorial	O
function	O
.	O
</s>
<s>
To	O
continue	O
the	O
induction	B-Algorithm
,	O
we	O
apply	O
integration	O
by	O
parts	O
to	O
the	O
error	B-Algorithm
term	I-Algorithm
:	O
</s>
<s>
Summing	O
from	O
to	O
and	O
substituting	O
this	O
for	O
the	O
lower	O
order	O
error	B-Algorithm
term	I-Algorithm
results	O
in	O
the	O
case	O
of	O
the	O
formula	O
,	O
</s>
<s>
In	O
this	O
way	O
we	O
get	O
a	O
proof	O
of	O
the	O
Euler	B-Algorithm
–	I-Algorithm
Maclaurin	I-Algorithm
summation	I-Algorithm
formula	I-Algorithm
which	O
can	O
be	O
formalized	O
by	O
mathematical	B-Algorithm
induction	I-Algorithm
,	O
in	O
which	O
the	O
induction	B-Algorithm
step	I-Algorithm
relies	O
on	O
integration	O
by	O
parts	O
and	O
on	O
identities	O
for	O
periodic	O
Bernoulli	O
functions	O
.	O
</s>
