December 2017 Archives

Finally On A Calculus Of Differences

My fellow students and I have spent much of our spare time this past year investigating the similarities between the calculus of functions and that of sequences, which we have defined for a sequence sn with the differential operator

  Δ sn = sn - sn-1

and the integral operator
  n
  Δ-1 sn = Σ si
  i = 1
where Σ is the summation sign, adopting the convention that terms with non-positive indices equate to zero.

We have thus far discovered how to differentiate and integrate monomial sequences, found product and quotient rules for differentiation, a rule of integration by parts and figured solutions to some familiar-looking differential equations, all of which bear a striking resemblance to their counterparts for functions. To conclude our investigation, we decided to try to find an analogue of Taylor's theorem for sequences.

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