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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) $$\sum a^x=\frac{a^x}{a1}.$$
3 $$\sum 2^x=2^x.$$
4 For $a\neq 2n\pi$ $$\sum\sin(ax)=\frac{\cos(a(x1/2))}{2\sin(a/2)}.$$
5 For $a\neq 2n\pi$ $$\sum\cos(ax)=\frac{\sin(a(x1/2))}{2\sin(a/2)}.$$
5 For $a\neq 2n\pi$ $$\sum\cos(ax)=\frac{\sin(a(x1/2))}{2\sin(a/2)}.$$
Table of Contents
Proofs: 1
Mentioned in:
Problems: 1
Solutions: 2
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References
Bibliography
 Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", AddisonWesley, 1994, 2nd Edition
 Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960
 Bool, George: "A Treatise on the Calculus of Finite Differences", Dover Publications, Inc., 1960