Sumk0M

Sumk0M is a compact notational form used in mathematics and computer science to denote the finite summation of a function f over an index k ranging from 0 to M. In standard notation, this is written as the sum Σ_{k=0}^{M} f(k). The expression sumk0M f thus denotes the same operation, commonly seen in informal notes, lecture slides, or programming comments for readability.

Definition and notation

For any function f defined on the integers 0 ≤ k ≤ M, sumk0M f(k) equals ∑_{k=0}^{M} f(k). If f is constant, say f(k) = c, the sum equals (M+1)c. If f(k) = k, the sum equals M(M+1)/2. More generally, sums of powers f(k) = k^p can be evaluated via Faulhaber-type formulas or polynomial expressions.

Properties

Sumk0M obeys linearity: sumk0M [f(k) + g(k)] = sumk0M f(k) + sumk0M g(k), and sumk0M [a f(k)] = a sumk0M f(k). It is sensitive to the upper limit M; extending M increases or alters the sum accordingly.

Applications

The notation is used in discrete mathematics, algorithm analysis, and probability, particularly when dealing with averages or total quantities over integer indices from 0 to M. It often appears in proofs, derivations, and code comments as a readable surrogate for Σ_{k=0}^{M} f(k).

History and usage

Sumk0M is not a formal standard symbol; it is an informal shorthand found in some texts, lecture notes, and programming contexts. Standard practice typically relies on the explicit Sigma notation.