Pi and the Euler number

When we write the Euler number e, what do we really mean? Since we know that e is the limit of the sequence (1+1/n)ⁿ, we can get an approximate answer rather easily by putting in a large number. For instance, n = 1000000 gives 2.7182805, which is pretty close to the popular value 2.718281828... Note however that this particular aproach is very sensitive to numerical error, there is a safer way to obtain the Euler number.

In the same way one can try to find the mysterious number pi. It can be proved that the sequence defined recursively by
(1)    a₁ = 1,
(2)    a_n+1 = a_n + (−1)ⁿ/(2n + 1), n = 1,2,3,...
is convergent and the limit is pi divided by four. The sumation actually goes 1 − 1/3 + 1/5 − 1/7 +... Finding the 100th term took some time (to write the appropriate program for my calculator), but eventually I obtained a₁₀₀ = 0.787873. This multiplied by four gives about 3.151, which is quite close to 3.14159265358979323.... If I had time, I could ask my calculator for a_1000000 and get π with better precision.

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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