Editing Summation (section)

=== General identities ===
: <math>\sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n) \quad</math> ([[distributivity]])<ref name="vpr">{{cite book
 | last1 = Varberg | first1 = Dale E.
 | last2 = Purcell | first2 = Edwin J.
 | last3 = Rigdon | first3 = Steven E.
 | title = Calculus
 | year = 2007
 | publisher = [[Pearson Prentice Hall]]
 | edition = 9th
 | isbn = 978-0131469686
 | page = 217
}}</ref>
: <math>\sum_{n=s}^t f(n) \pm \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left(f(n) \pm g(n)\right)\quad</math> ([[commutativity]] and [[associativity]])<ref name="vpr"/>
: <math>\sum_{n=s}^t f(n) = \sum_{n=s+p}^{t+p} f(n-p)\quad</math> (index shift)
: <math>\sum_{n\in B} f(n) = \sum_{m\in A} f(\sigma(m)), \quad</math> for a [[bijection]] {{mvar|σ}} from a finite set {{mvar|A}} onto a set {{mvar|B}} (index change); this generalizes the preceding formula.
: <math>\sum_{n=s}^t f(n) =\sum_{n=s}^j f(n) + \sum_{n=j+1}^t f(n)\quad</math> (splitting a sum, using [[associativity]])
: <math>\sum_{n=a}^{b}f(n)=\sum_{n=0}^{b}f(n)-\sum_{n=0}^{a-1}f(n)\quad</math> (a variant of the preceding formula)
: <math>\sum_{n=s}^t f(n) = \sum_{n=0}^{t-s} f(t-n)\quad</math> (the sum from the first term up to the last is equal to the sum from the last down to the first)
: <math>\sum_{n=0}^t f(n) = \sum_{n=0}^{t} f(t-n)\quad</math> (a particular case of the formula above)
: <math>\sum_{i=k_0}^{k_1}\sum_{j=l_0}^{l_1} a_{i,j} = \sum_{j=l_0}^{l_1}\sum_{i=k_0}^{k_1} a_{i,j}\quad</math> (commutativity and associativity, again)
: <math>\sum_{k\le j \le i\le n} a_{i,j} = \sum_{i=k}^n\sum_{j=k}^i a_{i,j} = \sum_{j=k}^n\sum_{i=j}^n a_{i,j} =
\sum_{j=0}^{n-k}\sum_{i=k}^{n-j} a_{i+j,i}\quad</math> (another application of commutativity and associativity)
: <math>\sum_{n=2s}^{2t+1} f(n) = \sum_{n=s}^t f(2n) + \sum_{n=s}^t f(2n+1)\quad</math> (splitting a sum into its [[parity (mathematics)|odd]] and [[parity (mathematics)|even]] parts, for even indexes)
: <math>\sum_{n=2s+1}^{2t} f(n) = \sum_{n=s+1}^t f(2n) + \sum_{n=s+1}^t f(2n-1)\quad</math> (splitting a sum into its odd and even parts, for odd indexes)
:<math>\biggl(\sum_{i=0}^{n} a_i\biggr) \biggl(\sum_{j=0}^{n} b_j\biggr)=\sum_{i=0}^n \sum_{j=0}^n a_ib_j \quad</math> ([[distributivity]])
: <math>\sum_{i=s}^m\sum_{j=t}^n {a_i}{c_j} = \biggl(\sum_{i=s}^m a_i\biggr) \biggl( \sum_{j=t}^n c_j \biggr)\quad</math> (distributivity allows factorization)
: <math>\sum_{n=s}^t \log_b f(n) = \log_b \prod_{n=s}^t f(n)\quad</math> (the [[logarithm]] of a product is the sum of the logarithms of the factors)
: <math>C^{\sum\limits_{n=s}^t f(n) } = \prod_{n=s}^t C^{f(n)}\quad</math> (the [[exponentiation|exponential]] of a sum is the product of the exponential of the summands)
: <math>\sum^{k}_{m = 0}\sum^{m}_{n = 0}f(m,n)=\sum^{k}_{m = 0}\sum^{k}_{n = m}f(n,m),\quad</math>for any function <math display="inline">f</math> from <math display="inline">\mathbb{Z}\times\mathbb{Z}</math>.