Proposition: Antidifferences of Some Functions

The following are examples of functions and their antidifferences. The examples show that the methods of difference calculus give us new and powerful methods to find closed formulas for many sums.

1 For $n\neq -1$ and the sum of falling factorial powers $$\sum x^{\underline{n}}=\frac{x^{\underline{n+1}}}{n+1}.$$

2 For $a\neq 1$ (in analogy to the formula for the sum of geometric progression) $$\sum a^x=\frac{a^x}{a-1}.$$

3 $$\sum 2^x=2^x.$$

4 For $a\neq 2n\pi$ $$\sum\sin(ax)=-\frac{\cos(a(x-1/2))}{2\sin(a/2)}.$$

5 For $a\neq 2n\pi$ $$\sum\cos(ax)=\frac{\sin(a(x-1/2))}{2\sin(a/2)}.$$

5 For $a\neq 2n\pi$ $$\sum\cos(ax)=\frac{\sin(a(x-1/2))}{2\sin(a/2)}.$$

Proofs: 1

Problems: 1
Solutions: 2

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  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