Summation Formula List

In mathematics, summation is the addition of a sequence of numbers. The result is a sum or total. The sum of a sequence of numbers is denoted with an enlarged capital Greek sigma symbol $\sum$ . Summation is used in mathematics to approximate definite integrals; describe statistical distributions and estimators; and denote combinatorial computations.

Terminology

File:Greek uc sigma.svg

The summation symbol

The sum of a sequence of numbers is denoted by:

\sum_{i = 1}^{n} a_{i} = a_{1} + a_{2} + a_{3} + \dots + a_{n - 1} + a_{n}

where i represents the index; a_i are the successive terms in the sum; 1 is the lower bound, and n is the upper bound. The index, i, is incremented by 1 for each successive term, stopping when i = n The numbers to be summed are called addends, or sometimes summands. The addends, represented by a_i, may be integers, rational numbers, real numbers, or complex numbers. ^[1]

Formulae

$\sum_{i = 1}^{n} 1 = n$
$\sum_{i = 1}^{n} c = n c$ for every constant $c$
$\sum_{i = 0}^{n} i = \sum_{i = 1}^{n} i = \frac{n (n + 1)}{2}$ (Sum of the simplest arithmetic progression, consisting of the n first natural numbers.)^[2]
$\sum_{i = 1}^{n} 2 i - 1 = n^{2}$ (Sum of first odd natural numbers)
$\sum_{i = 0}^{n} 2 i = n (n + 1)$ (Sum of first even natural numbers)
$\sum_{i = 1}^{n} \log i = \log n!$ (A sum of logarithms is the logarithm of the product)
$\sum_{i = 0}^{n} i^{2} = \frac{n (n + 1) (2 n + 1)}{6} = \frac{n^{3}}{3} + \frac{n^{2}}{2} + \frac{n}{6}$ (Sum of the first squares, see square pyramidal number.) ^[2]
$\sum_{i = 0}^{n} i^{3} = {(\sum_{i = 0}^{n} i)}^{2} = {(\frac{n (n + 1)}{2})}^{2} = \frac{n^{4}}{4} + \frac{n^{3}}{2} + \frac{n^{2}}{4}$ (Nicomachus's theorem) ^[2]
$\sum_{i = 0}^{n} i^{4} = \frac{n (n + 1) (2 n + 1) (3 n^{2} + 3 n - 1)}{30} = \frac{n^{5}}{5} + \frac{n^{4}}{2} + \frac{n^{3}}{3} - \frac{n}{30}$ ^[2]
$\sum_{i = 1}^{n} i^{5} = \frac{n^{2} (n + 1)^{2} (2 n^{2} + 2 n - 1)}{12}$ ^[2]
$\sum_{i = 1}^{n} i^{6} = \frac{n (n + 1) (2 n + 1) (3 n^{4} + 6 n^{3} - 3 n + 1)}{42}$ ^[2]
$\sum_{i = 1}^{n} i^{7} = \frac{n^{2} (n + 1)^{2} (3 n^{4} + 6 n^{3} - n^{2} - 4 n + 2)}{24}$ ^[2]
$\sum_{i = 1}^{n} i^{8} = \frac{n (n + 1) (2 n + 1) (5 n^{6} + 15 n^{5} + 5 n^{4} - 15 n^{3} - n^{2} - 9 n - 3)}{90}$ ^[2]
$\sum_{i = 1}^{n} i^{9} = \frac{n^{2} (n + 1)^{2} (2 n^{6} + 6 n^{5} + n^{4} - 8 n^{3} + n^{2} + 6 n - 3)}{20}$ ^[2]
$\sum_{i = 1}^{n} i^{10} = \frac{n (n + 1) (2 n + 1) (3 n^{8} + 12 n^{7} + 8 n^{6} - 18 n^{5} - 10 n^{4} + 24 n^{3} + 2 n^{2} - 15 n + 5)}{66}$ ^[2]
$\sum_{i = 1}^{n} 3 i^{2} - 3 i + 1 = n^{3}$ (exact cubic closed form)
$\sum_{i = 1}^{n} 4 i^{3} - 6 i^{2} + 4 i - 1 = n^{4}$ (exact quartic closed form)
$\sum_{i = 1}^{n} 5 i^{4} - 10 i^{3} + 10 i^{2} - 5 i + 1 = n^{5}$ (exact quintic closed form)
$\sum_{i = 1}^{n} 6 i^{5} - 15 i^{4} + 20 i^{3} - 15 i^{2} + 6 i - 1 = n^{6}$ (exact sextic closed form)
$\sum_{i = 1}^{n} 7 i^{6} - 21 i^{5} + 35 i^{4} - 35 i^{3} + 21 i^{2} - 7 i + 1 = n^{7}$ (exact septic closed form)
$\sum_{i = 1}^{n} 8 i^{7} - 28 i^{6} + 56 i^{5} - 70 i^{4} + 56 i^{3} - 28 i^{2} + 8 i - 1 = n^{8}$ (exact octic closed form)
$\sum_{i = 1}^{n} 9 i^{8} - 36 i^{7} + 84 i^{6} - 126 i^{5} + 126 i^{4} - 84 i^{3} + 36 i^{2} - 9 i + 1 = n^{9}$ (exact nonic closed form)
$\sum_{i = 1}^{n} 10 i^{9} - 45 i^{8} + 120 i^{7} - 210 i^{6} + 252 i^{5} - 210 i^{4} + 120 i^{3} - 45 i^{2} + 10 i - 1 = n^{10}$ (exact decic closed form)
$\sum_{i = 0}^{n - 1} a^{i} = \frac{1 - a^{n}}{1 - a}$ , $a \neq 1$ , (see geometric series)
$\sum_{i = 0}^{n - 1} \frac{1}{2^{i}} = 2 - \frac{1}{2^{n - 1}}$
$\sum_{i = 0}^{n - 1} i a^{i} = \frac{a - n a^{n} + (n - 1) a^{n + 1}}{(1 - a)^{2}}$ , $a \neq 1$ .
$\sum_{i = 0}^{n - 1} i 2^{i} = 2 + (n - 2) 2^{n}$
$\sum_{i = 0}^{n - 1} \frac{i}{2^{i}} = 2 - \frac{n + 1}{2^{n - 1}}$
$\begin{matrix} \sum_{i = 0}^{n - 1} (b + i d) a^{i} & = \frac{b - [b + (n - 1) d] a^{n}}{1 - a} + \frac{d a (1 - a^{n - 1})}{(1 - a)^{2}} \end{matrix}$ , $a \neq 1$ (see arithmetico-geometric series)
$\sum_{i = 0}^{n} (\binom{n}{i}) = 2^{n}$ (Gives the number of combinations in the binomial distribution)
$\sum_{i = 0}^{n} (\binom{n}{i}) p^{i} (1 - p)^{n - i} = 1$ , $0 \leq p \leq 1.$ (The binomial distribution)
$\sum_{k = 0}^{m} (\begin{matrix} n + k \\ n \end{matrix}) = (\begin{matrix} n + m + 1 \\ n + 1 \end{matrix})$
$\sum_{i = 1}^{n} i (\binom{n}{i}) = n (2^{n - 1})$
$\sum_{i = 0}^{n} \frac{(\binom{n}{i})}{i + 1} = \frac{2^{n + 1} - 1}{n + 1}$
$\sum_{i = k}^{n} (\binom{i}{k}) = (\binom{n + 1}{k + 1})$
$\sum_{i = 0}^{n} (\binom{n}{i}) a^{n - i} b^{i} = (a + b)^{n}$ , the binomial theorem
$\sum_{i = 0}^{n} i \cdot i! = (n + 1)! - 1$
$\sum_{i = 0}^{n} (\binom{m + i - 1}{i}) = (\binom{m + n}{n})$
$\sum_{i = 0}^{n} {(\binom{n}{i})}^{2} = (\binom{2 n}{n})$

Notes

↑ Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). "Chapter 2: Sums". Concrete Mathematics: A Foundation for Computer Science (2nd Edition). Addison-Wesley Professional.CS1 maint: Uses authors parameter (link) Search this book on
↑ ^2.00 ^2.01 ^2.02 ^2.03 ^2.04 ^2.05 ^2.06 ^2.07 ^2.08 ^2.09 W. H. Boyer (Editor), CRC Standard Math Tables, CRC Press, p 52, 1984

External links

"Summation". PlanetMath.

References

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[1] Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). "Chapter 2: Sums". Concrete Mathematics: A Foundation for Computer Science (2nd Edition). Addison-Wesley Professional.CS1 maint: Uses authors parameter (link) Search this book on

[CRC-2] 2.00 ^2.01 ^2.02 ^2.03 ^2.04 ^2.05 ^2.06 ^2.07 ^2.08 ^2.09 W. H. Boyer (Editor), CRC Standard Math Tables, CRC Press, p 52, 1984

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Summation Formula List

Contents

Terminology

Formulae

See also

Notes

External links

References

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