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 s_{n} with the differential operator
Δ s_{n} = s_{n} - s_{n-1}
and the integral operator
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.
and the integral operator
n | |||
Δ^{-1} s_{n} = | Σ | s_{i} | |
i = 1 |
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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