User:RLGoodwin/sandbox

This page is about the mathematics of finite summation. For the more elementary aspects of the topic, see Addition. For infinite summation, see Series (mathematics). For other uses, see RLGoodwin/sandbox (disambiguation).

Closed Form Summation Formulas

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 $\textstyle \sum$. Summation is used in mathematics to approximate definite integrals; describe statistical distributions and estimators; and denote combinatoric computations.

Terminology

The summation symbol

The sum of a sequence of numbers is denoted by:

$\sum _{i\mathop {=} 1}^{n}a_{i}=a_{1}+a_{2}+a_{3}+\cdots +a_{n-1}+a_{n}$

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. .

Measure theory notation

Using the notation from measure and integration theory, a sum can be expressed as a definite integral,

$\sum _{k\mathop {=} a}^{b}f(k)=\int _{[a,b]}f\,d\mu$

where $[a,b]$ is the subset of the integers from $a$ to $b$, and where $\mu$ is the counting measure.

Fundamental Theorem of Calculus

Let $f$ be a function that is continuous over the domain $a\leq x\leq b.$ Let $\{a<x_{1}<x_{2}<\cdots <x_{n-1}<b\}$ be a set of ordered numbers which partition the interval $(a,b)$ into $n$ equal subintervals each of length $\Delta x={\frac {(b-a)}{n}}.$ Let $S_{n}=\sum _{k=1}^{n}f(c_{k})\Delta x.$ Finally, let $F(x)$ be any integral of $f(x)\,\,dx$ such that

$F(x)=\int f(x)\,\,\,dx.$

Then, as $n\rightarrow \infty ,$ $\lim \sum f(c_{k})\Delta x=F(b)-F(a).$

Formulae

$\sum _{i=1}^{n}c=nc\quad$ for every constant c

$\sum _{i=0}^{n}i=\sum _{i=1}^{n}i={\frac {n(n+1)}{2}}\qquad$ (Sum of the simplest arithmetic progression, consisting of the n first natural numbers.)

$\sum _{i=1}^{n}2i-1=n^{2}\qquad$ (Sum of first odd natural numbers)

$\sum _{i=0}^{n}2i=n(n+1)\qquad$ (Sum of first even natural numbers)

$\sum _{i=1}^{n}\log i=\log n!\qquad$ (A sum of logarithms is the logarithm of the product)

$\sum _{i=0}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}={\frac {n^{3}}{3}}+{\frac {n^{2}}{2}}+{\frac {n}{6}}\qquad$ (Sum of the first squares, see square pyramidal number.)

$\sum _{i=0}^{n}i^{3}=\left(\sum _{i=0}^{n}i\right)^{2}=\left({\frac {n(n+1)}{2}}\right)^{2}={\frac {n^{4}}{4}}+{\frac {n^{3}}{2}}+{\frac {n^{2}}{4}}\qquad$ (Nicomachus's theorem)

$\sum _{i=0}^{n}i^{4}={\frac {n(n+1)(2n+1)(3n^{2}+3n-1)}{30}}={\frac {n^{5}}{5}}+{\frac {n^{4}}{2}}+{\frac {n^{3}}{3}}-{\frac {n}{30}}$

$\sum _{i=1}^{n}i^{5}={\frac {n^{2}(n+1)^{2}(2n^{2}+2n-1)}{12}}$

$\sum _{i=1}^{n}i^{6}={\frac {n(n+1)(2n+1)(3n^{4}+6n^{3}-3n+1)}{42}}$

$\sum _{i=1}^{n}i^{7}={\frac {n^{2}(n+1)^{2}(3n^{4}+6n^{3}-n^{2}-4n+2)}{24}}$

$\sum _{i=1}^{n}i^{8}={\frac {n(n+1)(2n+1)(5n^{6}+15n^{5}+5n^{4}-15n^{3}-n^{2}-9n-3)}{90}}$

$\sum _{i=1}^{n}i^{9}={\frac {n^{2}(n+1)^{2}(2n^{6}+6n^{5}+n^{4}-8n^{3}+n^{2}+6n-3)}{20}}$

$\sum _{i=1}^{n}i^{10}={\frac {n(n+1)(2n+1)(3n^{8}+12n^{7}+8n^{6}-18n^{5}-10n^{4}+24n^{3}+2n^{2}-15n+5)}{66}}$

$\sum _{i=0}^{n}i^{p}={\frac {(n+1)^{p+1}}{p+1}}+\sum _{k=1}^{p}{\frac {B_{k}}{p-k+1}}{p \choose k}(n+1)^{p-k+1},$ where $B_{k}$ denotes a Bernoulli number (see Faulhaber's formula).

$\sum _{i=1}^{n}3i^{2}-3i+1=n^{3}$ (exact cubic closed form)

$\sum _{i=1}^{n}4i^{3}-6i^{2}+4i-1=n^{4}$ (exact quartic closed form)

$\sum _{i=1}^{n}5i^{4}-10i^{3}+10i^{2}-5i+1=n^{5}$ (exact quintic closed form)

$\sum _{i=1}^{n}6i^{5}-15i^{4}+20i^{3}-15i^{2}+6i-1=n^{6}$ (exact sextic closed form)

$\sum _{i=1}^{n}7i^{6}-21i^{5}+35i^{4}-35i^{3}+21i^{2}-7i+1=n^{7}$ (exact septic closed form)

$\sum _{i=1}^{n}8i^{7}-28i^{6}+56i^{5}-70i^{4}+56i^{3}-28i^{2}+8i-1=n^{8}$ (exact octic closed form)

$\sum _{i=1}^{n}9i^{8}-36i^{7}+84i^{6}-126i^{5}+126i^{4}-84i^{3}+36i^{2}-9i+1=n^{9}$ (exact nonic closed form)

$\sum _{i=1}^{n}10i^{9}-45i^{8}+120i^{7}-210i^{6}+252i^{5}-210i^{4}+120i^{3}-45i^{2}+10i-1=n^{10}$ (exact decic closed form)

$\sum _{i=0}^{n-1}a^{i}={\frac {1-a^{n}}{1-a}}$, $a\neq 1$, (see geometric series)

$\sum _{i=0}^{n-1}{\frac {1}{2^{i}}}=2-{\frac {1}{2^{n-1}}}$

$\sum _{i=0}^{n-1}ia^{i}={\frac {a-na^{n}+(n-1)a^{n+1}}{(1-a)^{2}}}$, $a\neq 1$.

$\sum _{i=0}^{n-1}i2^{i}=2+(n-2)2^{n}$

$\sum _{i=0}^{n-1}{\frac {i}{2^{i}}}=2-{\frac {n+1}{2^{n-1}}}$

${\begin{aligned}\sum _{i=0}^{n-1}\left(b+id\right)a^{i}&={\frac {b-[b+(n-1)d]a^{n}}{1-a}}+{\frac {da(1-a^{n-1})}{(1-a)^{2}}}\end{aligned}}$, $a\neq 1$ (see arithmetico-geometric series)

$\sum _{i=0}^{n}{n \choose i}=2^{n}$ (Gives the number of combinations in the binomial distribution)

$\sum _{k=0}^{m}\left({\begin{array}{c}n+k\\n\\\end{array}}\right)=\left({\begin{array}{c}n+m+1\\n+1\\\end{array}}\right)$

$\sum _{i=1}^{n}i{n \choose i}=n(2^{n-1})$

$\sum _{i=0}^{n}{\frac {n \choose i}{i+1}}={\frac {2^{n+1}-1}{n+1}}$

$\sum _{i=k}^{n}{i \choose k}={n+1 \choose k+1}$

$\sum _{i=0}^{n}{n \choose i}a^{n-i}b^{i}=(a+b)^{n}$, the binomial theorem

$\sum _{i=0}^{n}i\cdot i!=(n+1)!-1$

$\sum _{i=0}^{n}{m+i-1 \choose i}={m+n \choose n}$

$\sum _{i=0}^{n}{n \choose i}^{2}={2n \choose n}$

External links

• Media related to RLGoodwin/sandbox at Wikimedia Commons
• "Summation". PlanetMath.

Category:Arithmetic Category:Mathematical notation Category:Addition