Definite and indefinite summation
For the sum
function in MATLAB^{®}, see sum
.
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sum(f
,i
) sum(f
,i = a .. b
) sum(f
,i = RootOf(p, x)
)
sum(f, i)
computes a symbolic antidifference
of f(i) with
respect to i.
sum(f, i = a..b)
tries to find a closed form
representation of the sum
.
sum
serves for simplifying symbolic sums
(the discrete analog of integration). It should not be
used for simply adding a finite number of terms: if a
and b
are
integers of type DOM_INT
, the call _plus(f $ i
= a..b)
gives the desired result, while sum(f,
i = a..b)
may return unevaluated. expand
may be used to sum such an unevaluated
finite sum. See Example 3.
sum(f, i)
computes the indefinite sum of f
with
respect to i
. This is an expression g
such
that f(i)
= g(i + 1)  g(i).
It is implicitly assumed that i
runs through
integers only.
sum(f, i = a..b)
computes the definite sum
with i
running from a
to b
.
If a
and b
are numbers,
then they must be integers.
If b  a
is a nonnegative integer, then the
explicit sum f(a)
+ f(a + 1) + … + f(b) is
returned, provided that this sum has no more than 1000 terms.
sum(f, i = RootOf(p, x))
computes the sum
with i
extending over all roots of the polynomial p
with
respect to x
.
If f
is a rational function of i
,
a closed form of the sum will be found.
See Example 2.
The system returns a symbolic call of sum
if
it cannot compute a closed form representation of the sum.
Infinite symbolic sums without symbolic parameters can be evaluated
numerically via float
or numeric::sum
. Cf. Example 4.
We compute some indefinite sums:
sum(1/(i^2  1), i)
sum(1/i/(i + 2)^2, i)
sum(binomial(n + i, i), i)
We compute some definite sums. Note that are valid boundaries:
sum(1/(i^2 + 21*i), i = 1..infinity)
sum(1/i, i = a .. a + 3)
expand(%)
We compute some sums over all roots of a polynomial:
sum(i^2, i = RootOf(x^3 + a*x^2 + b*x + c, x))
sum(1/(z + i), i = RootOf(x^4  y*x + 1, x))
sum
can compute finite sums if indefinite
summation succeeds:
sum(1/(i^2 + i), i = 1..100)
_plus
yields
the same result more quickly if the number of summands is small:
_plus(1/(i^2 + i) $ i = 1..100)
In such cases, sum
is much more efficient
than _plus
if
the number of summands is large:
sum(1/(i^2 + i), i = 1..10^30)
Finite sums for which no indefinite summation is possible are expanded if they have no more than 1000 terms:
sum(binomial(n, i), i = 0..4)
An application of expand
is
necessary to expand the binomials:
expand(%)
Finite sums with more than 1000 terms are not expanded:
sum(binomial(n, i), i = 0..1000)
You might use expand
here
to expand the sum and obtain a huge expression. If you really want
to do that, we recommend using _plus
directly.
However, if one of the boundaries is symbolic, then _plus
cannot
be used:
_plus(1/(i^2 + i) $ i = 1..n)
_plus(binomial(n, i) $ i = 0..n)
sum(1/(i^2 + i), i = 1..n), sum(binomial(n, i), i = 0..n)
The following infinite sum cannot be computed symbolically:
sum(ln(i)/i^5, i = 1..infinity)
We obtain a floatingpoint approximation via float
:
float(%)
Alternatively, the function numeric::sum
can be used directly. This
is usually much faster than applying float
, since it avoids the overhead of sum
attempting
to compute a symbolic representation:
numeric::sum(ln(i)/i^5, i = 1..infinity)

An arithmetical
expression depending on 

The summation index: an identifier or indexed identifier 

The boundaries: arithmetical expressions 

A polynomial of
type 

An indeterminate of 
The function sum
implements Abramov's algorithm
for rational expressions, Gosper's algorithm for hypergeometric expressions,
and Zeilberger's algorithm for the definite summation of holonomic
expressions.