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 Calculus \Cal"cu*lus\, n.; pl. Calculi. [L, calculus. See
     Calculate, and Calcule.]
     1. (Med.) Any solid concretion, formed in any part of the
        body, but most frequent in the organs that act as
        reservoirs, and in the passages connected with them; as,
        biliary calculi; urinary calculi, etc.
        [1913 Webster]
  
     2. (Math.) A method of computation; any process of reasoning
        by the use of symbols; any branch of mathematics that may
        involve calculation.
        [1913 Webster]

Summation

$S u m m a t i o n$

Capital-sigma notation

wikipedia: Summation#Capital-sigma_notation

Mathematical notation has a special representation for compactly representing summation of many similar terms: the summation symbol ∑ (U+2211), a large upright capital Sigma. This is defined thus:

$\sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +\dots+ x_{n-1} + x_n.$

The subscript gives the symbol for an index variable, i. Here, i represents the index of summation; m is the lower bound of summation, and n is the upper bound of summation. Here i = m under the summation symbol means that the index i starts out equal to m. Successive values of i are found by adding 1 to the previous value of i, stopping when i = n. We could as well have used k instead of i, as in

$\sum_{k=2}^6 k^2 = 2^2+3^2+4^2+5^2+6^2 = 90$ .

Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context, as in

$\sum x_i^2$

which is informally equivalent to

$\sum_{i=1}^n x_i^2$ .

One often sees generalizations of this notation in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. For example:

$\sum_{0\le k< 100} f(k)$

is the sum of f(k) over all (integer) k in the specified range,

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

is the sum of f(x) over all elements x in the set S, and

$\sum_{d|n}\;\mu(d)$

is the sum of μ(d) over all integers d dividing n.

(Remark: Although the name of the dummy variable does not matter (by definition), one usually uses letters from the middle of the alphabet (i through q) to denote integers, if there is a risk of confusion. For example, even if there should be no doubt about the interpretation, it could look slightly confusing to many mathematicians to see x instead of k in the above formulae involving k. See also wikipedia: typographical conventions in mathematical formulae.)

There are also ways to generalize the use of many sigma signs. For example,

$\sum_{\ell,\ell'}$

is the same as

$\sum_\ell\sum_{\ell'}.$

A similar notation is applied when it comes to finding multiplicative products; the same basic structure is used, with ∏, or the capital pi, replacing the ∑.

Capital pi notation

The product of a sequence of terms can be written with the product symbol, which derives from the capital letter Π (Pi) in the wikipedia: Greek alphabet. Unicode position U+220F (∏) contains a glyph for denoting such a product, distinct from U+03A0 (Π), the letter. The meaning of this notation is given by:

$\prod_{i=m}^{n} x_{i} = x_{m} \cdot x_{m+1} \cdot x_{m+2} \cdot \,\,\cdots\,\, \cdot x_{n-1} \cdot x_{n}.$

The subscript gives the symbol for a dummy variable (i in this case), called the "index of multiplication" together with its lower bound (m), whereas the superscript (here n) gives its upper bound. The lower and upper bound are expressions denoting integers. The factors of the product are obtained by taking the expression following the product operator, with successive integer values substituted for the index of multiplication, starting from the lower bound and incremented by 1 up to and including the upper bound. So, for example:

$\prod_{i=2}^{6} \left(1 + {1\over i}\right) = \left(1 + {1\over 2}\right) \cdot \left(1 + {1\over 3}\right) \cdot \left(1 + {1\over 4}\right) \cdot \left(1 + {1\over 5}\right) \cdot \left(1 + {1\over 6}\right) = {7\over 2}.$

In case m = n, the value of the product is the same as that of the single factor x_m. If m > n, the product is the empty product, with the value 1.

Infinite products

Main article: Infinite product

One may also consider products of infinitely many terms; these are called infinite products. Notationally, we would replace n above by the lemniscate (infinity symbol) ∞. In the reals, the product of such a series is defined as the limit of the product of the first n terms, as n grows without bound. That is, by definition,

$\prod_{i=m}^{\infty} x_{i} = \lim_{n\to\infty} \prod_{i=m}^{n} x_{i}.$