User:RLGoodwin/sandbox

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 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; ai 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 ai, 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 $$a \le x \le b. $$ 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)


 * $$\sum_{i=0}^{n-1} \frac{1}{2^i} = 2-\frac{1}{2^{n-1}}$$


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


 * $$\sum_{i=0}^{n-1} i 2^i = 2+(n-2)2^{n}$$


 * $$\sum_{i=0}^{n-1} \frac{i}{2^i} = 2-\frac{n+1}{2^{n-1}}$$


 * $$\begin {align}\sum_{i= 0}^{n-1} \left(b + i d\right) a^i &= \frac{b - [b+(n - 1)d] a^n}{1 - a}+\frac{da(1 - a^{n - 1})}{(1 - a)^2}\end {align}$$, $$a\neq 1$$ (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


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


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


 * $$\sum_{i=0}^n {n \choose i}^2 = {2n \choose n}$$