Indefinite summation

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Indefinite summation (for discrete variables) is the equivalent of indefinite integration (for continuous variables.)

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

$\Delta ^{-1}a(n)\equiv A(n)+C,\quad \Delta (A(n)+C)=A(n+1)-A(n)=a(n),\,$

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

Example

With

$a(n)=n\,$

we get

$\Delta ^{-1}a(n)=\Delta ^{-1}n={\frac {(n-1)n}{2}}+C=A(n)+C\,$

so that

$\Delta (A(n)+C)=\Delta {\bigg [}{\frac {(n-1)n}{2}}+C{\bigg ]}={\bigg [}{\frac {(n-1)n}{2}}+C{\bigg ]}_{n+1}-{\bigg [}{\frac {(n-1)n}{2}}+C{\bigg ]}_{n}={\frac {n(n+1)}{2}}-{\frac {(n-1)n}{2}}=n=a(n)\,$

See also

• Definite summation
• Indefinite summation

• Finite difference
  • Forward difference
  • Backward difference