π(π π β ) Γ π(gcd [ π ,
π
] = 1)
1 π 2
π 2
Γ =
Using the knowledge that two numbers must have a greatest common denominator, we know that the sum of the probabilities where π = 1, 2 ,3, β¦ , β must cover all pairings and so equal 1:
β
π 2
β
= 1
=1
Rearranged:
β
1 π 2
β
= 1
=1
1
=
1 π 2
β =1
β
Let this be result (2)
Eulerβs Solution to the Basel Problem
The result for the sum to infinity of π β2 is called the Basel problem. It was investigated in the 17 th and 18 th centuries until its exact solution was proved in 1734 by Euler. His initial proof (below) was criticised due to his assumption that what is true for a finite polynomial is also true for an infinite one. Despite this criticism, Euler later went on to prove that his result had been correct. Ever since, this result has been widely used and has been proved in countless other ways; however, Eulerβs is still the most elegant solution.
To commence the proof, Euler considered the Maclaurin series of sin - the expression that equals the function but represented in powers of . The Maclaurin series takes the general form:
2 β²β² (0) 2!
3 3 (0) 3!
β²(0) 1!
( ) = (0) +
+
+
+ β―
Applied to ( ) = sin :
β 2 sin(0) 2!
β 3 cos(0) 3!
4 sin(0) 4!
5 cos(0) 5!
cos(0) 1!
sin = sin(0) +
+
+
+
+
+ β―
3 3!
5 5!
sin = 0 + β 0 β
+ 0 +
+ β―
3 3!
5 5!
7 7!
2+1 (2π + 1)!
+ β―+ (β1)
sin = β
+
β
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