Proposition: Basic Calculations Involving Indefinite Sums

The indefinite sum is linear, i.e. for a complex-valued function $f:\mathbb C\to C$ and a constant $\lambda\in\mathbb C$ the following properties are fulfilled:

  1. $\sum \lambda f(x)=\lambda \sum f(x).$
  2. $\sum (f(x)\pm g(x))=\sum f(x)\pm \sum g(x).$

Moreover, for functions $f,g:\mathbb C\to C,$ the partial summation formula holds:

3. $\sum g(x)\Delta f(x)=f(x)g(x)-\sum f(x+1)\Delta g(x)$


Proofs: 1

Proofs: 1

Thank you to the contributors under CC BY-SA 4.0!




  1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition
  2. Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960
  3. Bool, George: "A Treatise on the Calculus of Finite Differences", Dover Publications, Inc., 1960