The Other Calculus: a Survey of the Calculus of Finite Differences
The Calculus of Finite Differences is a fascinating field of mathematics which is easily accessible to an undergraduate mathematics major. This paper attempts to present a broad survey of results from this field which show its richness, without appealing to deep results in analysis. Topics covered include differences, sums, factorial powers, approximation, and difference equations. An emphasis is given to the powerful Euler-Maclaurin sum formula because of its ability to connect differential and difference calculus and provide insight into asymptotic relationships. The paper includes comments on each section which point to deeper studies for those interested.
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