Summation

Summation is the Addition of a set of numbers; the result is their '''sum'''. For evaluation of sums in Closed Form see Evaluating Sums . The "numbers" to be summed may be Natural Number s, Complex Number s, Matrices , or still more complicated objects. An infinite sum is a subtle procedure known as a Series .

NOTATION

The sum of 1, 2, and 4 is 1 + 2 + 4 = 7.
Since addition is Associative , it does not matter whether we interpret "1 + 2 + 4" as (1 + 2) + 4 or as 1 + (2 + 4); the result is the same, so parentheses are usually omitted in a sum. Addition is also Commutative , so the order in which the numbers written does not affect its sum.

If a sum has too many terms to write them all out individually, the sum may be written with an Ellipsis to mark out the missing terms.
Thus, the sum of all the Natural Number s from 1 to 100 is 1 + 2 + … + 99 + 100 = 5050.

Sums can be represented by the summation symbol, a capital Sigma . This is defined as:
:

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

The subscript gives the symbol for a Dummy Variable , ''i''. Here, ''i'' represents the index of summation; ''m'' is the '''lower bound of summation''', and ''n'' is the '''upper bound of summation'''.
So, for example:
:

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

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

:where

B_k

is the ''k''th Bernoulli Number .

$\sum_{i=m}^n x^i = rac{x^{n+1}-x^{m}}{x-1}$ (see Geometric Series )

$\sum_{i=0}^n x^i = rac{1-x^{n+1}}{1-x}$ (special case of the above where ''m'' = 0)

$\sum_{i=0}^n {n \choose i} = 2^n$ (see Binomial Coefficient )

$\sum_{i=0}^{n-1} {i \choose k} = {n \choose k+1}$

$\left(\sum_i a_i
ight)\left(\sum_j b_j
ight) = \sum_i\sum_j a_ib_j$

$\left(\sum_i a_i
ight)^2 = 2\sum_i\sum_{j$

GROWTH RATES

The following are useful Approximation s (using Theta Notation ):

$\sum_{i=1}^n i^c = \Theta(n^{c+1})$ for real ''c'' greater than -1

$\sum_{i=1}^n rac{1}{i} = \Theta(\log n)$

$\sum_{i=1}^n c^i = \Theta(c^n)$ for real ''c'' greater than 1

$\sum_{i=1}^n \log(i)^c = \Theta(n \cdot \log(n)^{c})$ for Nonnegative real ''c''

$\sum_{i=1}^n \log(i)^c \cdot i^d = \Theta(n^{d+1} \cdot \log(n)^{c})$ for nonnegative real ''c'', ''d''

$\sum_{i=1}^n \log(i)^c \cdot i^d \cdot b^i = \Theta (n^d \cdot \log(n)^c \cdot b^n)$ for nonnegative real ''b'' > 1, ''c'', ''d''

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Summation