Since the integrand is continuous at that point and non-negative elsewhere, the integral from 0 to 1 must be strictly positive.

It remains to show that the integral in fact evaluates to the desired quantity:

 

(See polynomial long division.)

Quick upper and lower bounds
In , it is pointed out that if 1 is substituted for  in the denominator, one gets a lower bound on the integral, and if 0 is substituted for  in the denominator, one gets an upper bound:

Thus we have

hence 3.1412 <  < 3.1421 in decimal expansion. The bounds deviate by less than 0.015% from . See also .

Proof that 355/113 exceeds 
As discussed in , the well-known Diophantine approximation and far better upper estimate  for  follows from the relation

where the first six digitsAnd the Kenyan economy as a whole has been recovering, growing 4% in 2010.

Kenya’s wildlife attracts many visitors to the country’s national parks and game reserves. Its sandy beaches and coral reefs are also a draw for foreign visitors. Tourism is therefore of key importance to the economy.

M-Pesa

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