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

The sum of a sequence of numbers is denoted by:

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

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. .^[1]

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

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

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

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

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

Then, as ${\displaystyle n\rightarrow \infty ,}$ ${\displaystyle \lim \sum f(c_{k})\Delta x=F(b)-F(a).}$^[2]

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

${\displaystyle \sum _{i=0}^{n}i=\sum _{i=1}^{n}i={\frac {n(n+1)}{2}}\qquad }$ (Sum of the simplest arithmetic progression, consisting of the n first natural numbers.)^[3]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Category:Arithmetic Category:Mathematical notation Category:Addition