When we write the Euler number *e*, what do we really mean? Since we
know that *e* is the limit of the sequence
*n*)^{n},*n* = 1000000

In the same way one can try to find the mysterious number pi. It can be proved
that the sequence defined recursively by

(1) *a*_{1} = 1,

(2)
*a*_{n+1} = *a*_{n} + (−1)^{n}/(2*n* + 1),*n* = 1,2,3,...

is convergent and the limit is pi divided by four. The sumation actually goes
*a*_{100} = 0.787873.*a*_{1000000}

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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