Summation Formula List

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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

The summation symbol

The sum of a sequence of numbers is denoted by:

$\sum _{i=1}^n{a}_i=a_1+a_2+a_3+\cdots +a_{n-1}+a_n$

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.

Formulae

1. $\sum _{i=1}^{n}1=n$
2. $\sum _{i=1}^{n}c=nc$ for every constant c
3. $\sum _{i=0}^{n}i=\sum _{i=1}^{n}i=\frac{n(n+1)}{2}$ (Sum of the simplest arithmetic progression, consisting of the n first natural numbers.)
4. $\sum _{i=1}^{n}2i-1=n^2$ (Sum of first odd natural numbers)
5. $\sum _{i=0}^{n}2i=n(n+1)$ (Sum of first even natural numbers)
6. $\sum _{i=1}^n\log i=\log n!$ (A sum of logarithms is the logarithm of the product)
7. $\sum _{i=0}^n{i}^2=\frac{n(n+1)(2n+1)}{6}=\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}$ (Sum of the first squares, see square pyramidal number.)
8. $\sum _{i=0}^n{i}^3={\left(\sum _{i=0}^{n}i\right)}^2={\left(\frac{n(n+1)}{2}\right)}^2=\frac{n^4}{4}+\frac{n^3}{2}+\frac{n^2}{4}$ (Nicomachus's theorem)
9. $\sum _{i=0}^n{i}^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}=\frac{n^5}{5}+\frac{n^4}{2}+\frac{n^3}{3}-\frac{n}{30}$
10. $\sum _{i=1}^n{i}^5=\frac{n^2(n+1{)}^2(2n^2+2n-1)}{12}$
11. $\sum _{i=1}^n{i}^6=\frac{n(n+1)(2n+1)(3n^4+6n^3-3n+1)}{42}$
12. $\sum _{i=1}^n{i}^7=\frac{n^2(n+1{)}^2(3n^4+6n^3-n^2-4n+2)}{24}$
13. $\sum _{i=1}^n{i}^8=\frac{n(n+1)(2n+1)(5n^6+15n^5+5n^4-15n^3-n^2-9n-3)}{90}$
14. $\sum _{i=1}^n{i}^9=\frac{n^2(n+1{)}^2(2n^6+6n^5+n^4-8n^3+n^2+6n-3)}{20}$
15. $\sum _{i=1}^n{i}^{10}=\frac{n(n+1)(2n+1)(3n^8+12n^7+8n^6-18n^5-10n^4+24n^3+2n^2-15n+5)}{66}$
16. $\sum _{i=1}^{n}3i^2-3i+1=n^3$ (exact cubic closed form)
17. $\sum _{i=1}^{n}4i^3-6i^2+4i-1=n^4$ (exact quartic closed form)
18. $\sum _{i=1}^{n}5i^4-10i^3+10i^2-5i+1=n^5$ (exact quintic closed form)
19. $\sum _{i=1}^{n}6i^5-15i^4+20i^3-15i^2+6i-1=n^6$ (exact sextic closed form)
20. $\sum _{i=1}^{n}7i^6-21i^5+35i^4-35i^3+21i^2-7i+1=n^7$ (exact septic closed form)
21. $\sum _{i=1}^{n}8i^7-28i^6+56i^5-70i^4+56i^3-28i^2+8i-1=n^8$ (exact octic closed form)
22. $\sum _{i=1}^{n}9i^8-36i^7+84i^6-126i^5+126i^4-84i^3+36i^2-9i+1=n^9$ (exact nonic closed form)
23. $\sum _{i=1}^{n}10i^9-45i^8+120i^7-210i^6+252i^5-210i^4+120i^3-45i^2+10i-1=n^{10}$ (exact decic closed form)
24. $\sum _{i=0}^{n-1}{a}^i=\frac{1-a^n}{1-a}$, $a\ne 1$, (see geometric series)
25. $\sum _{i=0}^{n-1}\frac{1}{2^i}=2-\frac{1}{2^{n-1}}$
26. $\sum _{i=0}^{n-1}i{a}^i=\frac{a-n{a}^n+(n-1)a^{n+1}}{(1-a{)}^2}$, $a\ne 1$.
27. $\sum _{i=0}^{n-1}i{2}^i=2+(n-2)2^n$
28. $\sum _{i=0}^{n-1}\frac{i}{2^i}=2-\frac{n+1}{2^{n-1}}$
29. $\begin{array}{rl}\sum _{i=0}^{n-1}(b+id)a^i& =\frac{b-[b+(n-1)d]a^n}{1-a}+\frac{da(1-a^{n-1})}{(1-a{)}^2}\end{array}$, $a\ne 1$ (see arithmetico-geometric series)
30. $\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})=2^n$ (Gives the number of combinations in the binomial distribution)
31. $\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})p^i(1-p{)}^{n-i}=1$, $0\le p\le 1.$ (The binomial distribution)
32. $\sum _{k=0}^m\left(\begin{array}{c}n+k\\ n\\ \end{array}\right)=\left(\begin{array}{c}n+m+1\\ n+1\\ \end{array}\right)$
33. $\sum _{i=1}^{n}i(\genfrac{}{}{0ex}{}{n}{i})=n(2^{n-1})$
34. $\sum _{i=0}^n\frac{(\genfrac{}{}{0ex}{}{n}{i})}{i+1}=\frac{2^{n+1}-1}{n+1}$
35. $\sum _{i=k}^n(\genfrac{}{}{0ex}{}{i}{k})=(\genfrac{}{}{0ex}{}{n+1}{k+1})$
36. $\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})a^{n-i}{b}^i=(a+b{)}^n$, the binomial theorem
37. $\sum _{i=0}^{n}i\cdot i!=(n+1)!-1$
38. $\sum _{i=0}^n(\genfrac{}{}{0ex}{}{m+i-1}{i})=(\genfrac{}{}{0ex}{}{m+n}{n})$
39. $\sum _{i=0}^n{(\genfrac{}{}{0ex}{}{n}{i})}^2=(\genfrac{}{}{0ex}{}{2n}{n})$

See also

• Series (mathematics)
• List of integrals
• Taylor series
• Binomial theorem
• Gregory's series
• On-Line Encyclopedia of Integer Sequences

External links

• "Summation". PlanetMath.

References

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