Definite summation

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Definite summation (for discrete variables) is the equivalent of definite integration (for continuous variables.)

The definite summation of a discrete function (or sequence) $\scriptstyle a_{n}\,$ is

$\sum _{n=n_{\min }}^{n_{\max }}a(n)={\bigg [}\Delta ^{-1}a(n){\bigg ]}_{n_{\min }}^{n_{\max }+1}={\bigg [}A(n)+C{\bigg ]}_{n_{\min }}^{n_{\max }+1}=A(n_{\max }+1)-A(n_{\min }),\quad \Delta (A(n)+C)=A(n+1)-A(n)=a(n),\,$

where $\scriptstyle \sum \,$ is the definite summation operator, $\scriptstyle \Delta ^{-1}\,$ is the indefinite summation operator, $\scriptstyle \Delta \,$ is the forward difference operator and $\scriptstyle C\,$ if the indefinite summation constant.

See also

• Definite summation
• Indefinite summation

• Finite difference
  • Forward difference
  • Backward difference