Fungrim entry: 044e42

Symbol: Sum —

\sum_{n} f(n)

— Sum

Sum(S), rendered as

\sum S

, gives the sum of the elements of the set

S

. The sum is required to be absolutely convergent.

Sum(f(n), For(n, a, b)), rendered as

\sum_{n=a}^{b} f(n)

, gives the sum of

f(n)

for integers

n

from

a

b

, where

a

and

b

should be integers or

-\infty

\infty

. If

a < b

, the sum is empty. The sum

\sum_{n=0}^{\infty} f(n)

is interpreted as

\lim_{N \to \infty} \sum_{n=0}^{N} f(n)

and can be conditionally convergent.

Sum(f(n), For(n, a, b), P(n)), rendered as

\sum_{\textstyle{n=a \atop P(n)}}^{b} f(n)

, is as above but only terms satisfying the predicate

P(n)

are included.

Sum(f(x), ForElement(x, S)), rendered as

\sum_{x \in S} f(x)

, gives the sum of

f(x)

for all

x

in the set

S

. The sum is required to be absolutely convergent.

Sum(f(x), ForElement(x, S), P(x)), rendered as

\sum_{\textstyle{x \in S \atop P(x)}} f(x)

, gives the sum of

f(x)

for all

x

in the set

S

and satisfying the predicate

P(x)

. The sum is required to be absolutely convergent.

Sum(f(x), For(x), P(x)), rendered as

\sum_{P(x)} f(x)

, gives the sum of

f(x)

for all

x

satisfying the predicate

P(x)

. The predicate

P(x)

should define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. The sum is required to be absolutely convergent.

The empty sum is 0. The sum can range over an uncountable number of terms, as long as only countably many terms are nonzero.

The special expression For(x), For(x, a, b) or ForElement(x) declares x as a locally bound variable within the scope of the arguments to this operator. Similarly For(Tuple(x, y)), ForElement(Tuple(x, y), S) etc. defines multiple locally bound variables.

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence

Source code for this entry:

Entry(ID("044e42"),
    SymbolDefinition(Sum, Sum(f(n), For(n)), "Sum"),
    Description(SourceForm(Sum(S)), ", rendered as ", Sum(S), ", gives the sum of the elements of the set", S, ". The sum is required to be absolutely convergent."),
    Description(SourceForm(Sum(f(n), For(n, a, b))), ", rendered as ", Sum(f(n), For(n, a, b)), ", gives the sum of", f(n), "for integers", n, "from", a, "to", b, ", where", a, "and", b, "should be integers or", Neg(Infinity), "or", Infinity, ". ", "If", Less(a, b), ", the sum is empty. The sum", Sum(f(n), For(n, 0, Infinity)), "is interpreted as", SequenceLimit(Sum(f(n), For(n, 0, N)), For(N, Infinity)), "and can be conditionally convergent."),
    Description(SourceForm(Sum(f(n), For(n, a, b), P(n))), ", rendered as ", Sum(f(n), For(n, a, b), P(n)), ", is as above but", "only terms satisfying the predicate", P(n), "are included."),
    Description(SourceForm(Sum(f(x), ForElement(x, S))), ", rendered as ", Sum(f(x), ForElement(x, S)), ", gives the sum of", f(x), "for all", x, "in the set", S, ". The sum is required to be absolutely convergent."),
    Description(SourceForm(Sum(f(x), ForElement(x, S), P(x))), ", rendered as ", Sum(f(x), ForElement(x, S), P(x)), ", gives the sum of", f(x), "for all", x, "in the set", S, "and satisfying the predicate", P(x), ". The sum is required to be absolutely convergent."),
    Description(SourceForm(Sum(f(x), For(x), P(x))), ", rendered as ", Sum(f(x), For(x), P(x)), ", gives the sum of", f(x), "for all", x, "satisfying the predicate", P(x), ". ", "The predicate", P(x), "should define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "The sum is required to be absolutely convergent."),
    Description("The empty sum is", 0, ". The sum can range over an uncountable number of terms, as long as only countably many terms are nonzero."),
    Description("The special expression", SourceForm(For(x)), ", ", SourceForm(For(x, a, b)), " or", SourceForm(ForElement(x)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "Similarly", SourceForm(For(Tuple(x, y))), ", ", SourceForm(ForElement(Tuple(x, y), S)), "etc. defines multiple locally bound variables."))

Topics using this entry

Operators