Summation

Summation is the operation of adding a sequence of numbers. In mathematics it is denoted by the Greek letter sigma, ∑, and is used to form finite sums and infinite series. For a sequence a_i indexed by i in a range m ≤ i ≤ n, the finite sum is S = ∑_{i=m}^n a_i, which equals a_m + a_{m+1} + ... + a_n. The intermediate totals S_k = ∑_{i=m}^k a_i are called partial sums.

An infinite series is the limit of such finite sums as the upper index grows without bound: ∑_{i=m}^∞ a_i = lim_{k→∞} ∑_{i=m}^k a_i, when the limit exists. If it does, the series is convergent and its sum is that limit; if not, the series diverges. A series is said to converge absolutely if ∑ |a_i| converges; absolute convergence implies convergence.

Examples: the geometric series ∑_{i=0}^∞ ar^i has sum a/(1−r) when |r|<1; the harmonic series ∑_{i=1}^∞ 1/i diverges. The finite sum ∑_{i=1}^n i equals n(n+1)/2.

Properties: summation is linear, meaning ∑ (c a_i) = c ∑ a_i and ∑ (a_i + b_i) = ∑ a_i + ∑ b_i when the sums exist. Summation is a fundamental tool in analysis and number theory and serves as the discrete counterpart to integration.