This is the type of series that is used by calculators to accurately approximate trigonometric ratios. Euler then considered a finite th degree polynomial, p(x) , with two properties:
1. π( ) has non zero roots :
,
, β¦ , π
2. π( ) =
Such a polynomial can take the form:
1
2
( ) = (1 β
) (1 β
) Γ β¦Γ (1 β
)
Euler continued:
2 3!
4 5!
6 7!
sin
= 1 β
+
β
+ β―
Knowing that the roots, where ( ) = 0 , of sin occur at integer multiples of π (if you have remembered to put your calculator into radians) allows us to represent the above series as follows:
sin
π
π
2π
2π
= (1 β
) (1 +
) (1 β
) (1 +
) Γ β¦
2 π 2
2 4π 2
2 9π 2
sin
= [1 β
] [1 β
] [1 β
] Γ β¦
sinπ₯ π₯
is an infinite series. That it can be represented like a finite polynomial was Eulerβs underlying assumption in this proof. It was the subsequent definitive proof of this element using the Weierstrauss Factorisation Theorem that led to the result being widely accepted as correct.
Expanding the above product gives:
sin
1 π 2
1 4π 2
1 9π 2
= 1 β 2 (
+
+
+ β―) + β―
2 3!
4 5!
6 7!
1 π 2
1 4π 2
1 9π 2
+ β― = 1 β 2 (
1 β
+
β
+
+
+ β―) + β―
Simplifying both sides to compare the 2 terms:
2 3!
1 π 2
1 4π 2
1 9π 2
= β 2 (
β
+
+
+ β―)
1 6
1 π 2
1 4π 2
1 9π 2
=
+
+
+ β―
π 2 6
1 2 2
1 3 2
1 4 2
= 1 +
+
+
+ β―
Thus:
β
π
π
β
=
π=
26
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