SUMMATION

Summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total.

For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses; summation of the first 100 natural numbers may be written as 1 + 2 + 3 + 4 + ⋅⋅⋅ + 99 + 100.

Summation can be described using the symbol Σ, the capital Greek letter sigma.

The sum of the first n natural integers can be denoted as ^{n}Σ_{i=1} i.

^{n}Σ_{i=m} a_{i} = a_{m} + a_{m+1} + a_{m+2} + ... + a_{n-1} + a_{n}; where i is the index of summation; a_{i} is an indexed variable representing each term
of the sum; m is the lower bound of summation, and n is the upper bound of summation; the index starts from m and is incremented by one for each successive term, stopping when i = n.

1^{2}+2^{2}+3^{2}+...+100^{2} = ^{100}Σ_{k=1} k^{2}

{a_{i} | a_{i} ∈ ℝ, i ∈ {1,...,N}}, ^{N}Σ_{i=1} a_{i} = a_{1} + a_{2} + a_{3} + a_{N}

^{N}Σ_{i=1} a_{i} = ^{N}Σ_{k=1} a_{k}

^{6}Σ_{k=3} k^{3} = 3^{3} + 4^{3} + 5^{3} + 6^{3}

^{6}Σ_{i=3} i^{3} = 3^{3} + 4^{3} + 5^{3} + 6^{3}

^{10}Σ_{k=1} 1/k = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10

^{N}Σ_{k=1} c ⋅ a_{k} = c ⋅ ^{N}Σ_{k=1} a_{k} (c ∈ ℝ)

c ⋅ a_{1} + c ⋅ a_{2} + ... + c ⋅ a_{n} = c ⋅ (a_{1} + a_{2} + ... + a_{n})

^{N}Σ_{k=1} a_{k} = ^{M}Σ_{k=1} a_{k} + ^{N}Σ_{M+1} a_{k}, M,N ∈ ℕ | M < N

^{N}Σ_{k=1} (a_{k} + b_{k}) = ^{N}Σ_{k=1} a_{k} + ^{N}Σ_{k=1} b_{k}