When we write the Euler number e, what do we really mean? Since we
know that e is the limit of the sequence
In the same way one can try to find the mysterious number pi. It can be proved
that the sequence defined recursively by
(1)
(2)
is convergent and the limit is pi divided by four. The sumation actually goes
Before computers came around and people had to do calculations by hand, expressing important numbers using sequences has been an important topic. A lot of effort went to finding not just any sequence, but a sequence that converges really fast and requires the least number of operations to yield the required precision. This is still a viable field, finding applications in computers, since they use approximations to determine pretty much all functions.
Bisection method and Newton
method
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