∑ (nary summation)
Computes the sum of all elements in a container or in a sequence. The elements may be numbers, vectors, or matrices.
Syntax

∑(X)

X
is a vector, matrix, list, or set


∑(expr, var, a, b)

expr
is an expression in one variablevar

var
is the variable inexpr

a
is the lower bound 
b
is the upper bound

Description
Summing a container
If X
is a container, then ∑(X)
is the sum of all elements in X
.
The following container types are supported:

vectors

the result is the sum of all components


matrices

the result is the sum of all entries


lists containing numbers

the result is the sum of all numbers


lists containing vectors

the result is the vector sum of all vectors


lists containing matrices

the result is the matrix sum of all matrices


sets containing numbers

the result is the sum of all numbers


sets containing vectors

the result is the vector sum of all vectors


sets containing matrices

the result is the matrix sum of all matrices

Typewise, the summands may be either real or complex, and the result is complex iff at least one of the summands is complex.
Summing a sequence
∑(expr, var, a, b)
computes the sum of the expression expr
in one variable var
as var
takes all integer values from a
to b
(inclusively). This precisely implements the sigma notation for nary sums.
expr
must return a number, a vector, or a matrix.
Notes
These note apply to both modes of operation: summing a container and summing a sequence.
The empty sum is 0
. Notice that the sum of an empty collection of vectors or matrices is also equal to the scalar 0
, because the system cannot tell that the collection indeed is an empty collection of vectors or matrices. And even if it could, it wouldn’t know the dimension of the vectors or the size of the matrices.
When summing a collection of objects, the objects must be of the same kind (numbers, vectors, or matrices). If they are vectors or matrices, they must all be of the same dimension or size, respectively.
Notice that ∑
is a function, not a prefix operator. Hence, to sum a container X
you must write ∑(X)
– ∑X
would be a syntax error.
Examples
Summing a container
∑(❨5, 2, 3, 0, −1, 4❩)
13
∑(IdentityMatrix(100))
100
∑(SequenceList(100))
5050
∑('(❨6, 1, 0❩, ❨2, 1, 3❩, ❨0, −2, 1❩, ❨1, 1, −2❩, ❨0, −3, 1❩, ❨−2, 4, 5❩))
⎛7⎞ e⎜2⎟ ⎝8⎠
∑('(❨❨5, 1❩, ❨3, −1❩❩, ❨❨2, −i❩, ❨i, 1❩❩, ❨❨5, 0❩, ❨−1, 2⋅i❩❩))
⎛ 12 1 − i⎞ ⎝2 + i 2⋅i ⎠
Summing a sequence
∑(k, k, 1, 100)
5050
∑(k^2, k, 1, 100)
338350
∑(1/k!, k, 0, 100)
2.71828182846 (=e)
∑(2^k⋅k!^2/(2⋅k + 1)!, k, 0, 100)
1.57079632679 (=π/2)
∑(❨1, k, k^2, k^3, k^4❩, k, 1, 100)
⎛ 100 ⎞ ⎜ 5050 ⎟ e⎜ 338350 ⎟ ⎜ 25502500 ⎟ ⎝2050333330⎠
∑(1/(1 + Fibonacci(2⋅n + 1)), n, 0, 100)
1.11803398875 (=√5/2)
∑(1/prime(n)^2, n, 1, 1000000)
0.452247416352