finite calculus

May 12th, 2015
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Finite Calculus:A Tutorial for Solving Nasty SumsDavid GleichJanuary 17, 2005AbstractIn this tutorial, I will first explain the need for finite calculus using an example sumI think is difficult to solve. Next, I will show where this sum actually occurs and why itis important. Following that, I will present all the mathematics behind finite calculusand a series of theorems to make it helpful before concluding with a set of examples toshow that it really is useful.Contents1 How to EvaluatePn2x=1 x ?22 The Computational Cost of Gaussian Elimination33 The3.13.23.34578Finite CalculusDiscrete Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The Indefinite Sum and the Discrete Anti-Derivative . . . . . . . . . . .Helpful Finite Calculus Theorems . . . . . . . . . . . . . . . . . . . . . .4 Making Finite Calculus Useful:Stirling and His Numbers104.1 Stirling Numbers (of the Second Kind) . . . . . . . . . . . . . . . . . . . 124.2 Proving the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Computing Stirling Numbers . . . . . . . . . . . . . . . . . . . . . . . . 155 Examples16Pn Galore...2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.1xx=15.2 A Double Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Average Codeword Length . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Conclusion1811PnHow to Evaluatex=1 x2

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