Euler-Maclaurin Summation Formula

​
summation function
-k
e
3
k
2
sin
(k)
1
log(k+1)
terms in series
2
The Euler–Maclaurin summation formula transforms a sum over the values of a function to a sum over the derivatives of a function. Its main advantage is that the new sum usually converges much more quickly than original. One must be careful, though, as the error term may not go to zero as more and more terms in the series are taken, as is shown by
ln(1+k)
.
The blue line represents the value of the sum to
n
terms in the series. The red line represents the progression of the sum after
n
terms. The green line represents the progression of the Euler–Maclaurin sum after
n
terms.

Details

The Euler–Maclaurin formula reads
N
∑
k=0
f(k)=
N
∫
0
f(k)dk+
f(N)+f(0)
2
+
m
∑
k=1
B
2k
(2k)!
(
(2k-1)
f
(N)-
(2k-1)
f
(0))+
R
m
,where
B
n
is the
th
n
Bernoulli number and
R
m
is a remainder term. The power of the Euler–Maclaurin summation formula lies in how quickly the new sum converges, even if the error term is unbounded.

External Links

Euler–Maclaurin Integration Formulas (Wolfram MathWorld)

Permanent Citation

Sam Nicoll
​
​"Euler-Maclaurin Summation Formula"​
​http://demonstrations.wolfram.com/EulerMaclaurinSummationFormula/​
​Wolfram Demonstrations Project​
​Published: January 23, 2012