∑ (n-ary 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 variable var

  • var is the variable in expr

  • 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 n-ary 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

See also

• sum (synonym)

• ∏ (product)