There is a remarkable expression for the number as an infinite product. Starting with definite integral
, we derive it as follows:
By
That is,
And so,
By repeated application of (1) we have the following values for :
For even ,
Similarly, for odd ,
i.e.,
Since for we have
It means
or,
Hence,
Moreover, we have
so that
And,
gives
Substituting (5) and (6) into (4) yields
Since by Squeeze Theorem for Sequences,
Consequently,
i.e.,
This is Wallis’ product representation for named after John Wallis who discovered it in 1665.
Maxima knows Wallis’ Pi:
Fig. 1
So does Mathematica:
Fig. 2
Its convergence to is illustrated in Fig. 3:
Fig. 3
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