The interest of my fellow students and I has been somewhat piqued of late by a curious similarity of the relationship between sequences and series to that between the derivatives and integrals of functions. Specifically, for a function f taking a non-negative argument x, we have
and for a sequence s having terms
s_{1}, s_{2}, s_{3}, ...
we can define a series S with terms
where Σ is the summation sign, from which we can recover the terms of the sequence with
s_{n} = S_{n} - S_{n-1}
using the convention thatS_{0} equals zero.
This similarity rather set us to wondering whether we could employ the language of calculus to reason about sequences and series.
x | |||
F(x) = | ∫ | f(x) dx | |
0 | |||
f(x) = | d | F(x) | |
dx |
and for a sequence s having terms
we can define a series S with terms
n | |||
S_{n} = s_{1} + s_{2} + s_{3} + ... + s_{n} = | Σ | s_{i} | |
i = 1 |
using the convention that
This similarity rather set us to wondering whether we could employ the language of calculus to reason about sequences and series.
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