π/4 = 1 - 1/3 + 1/5 - 1/7 + ...

In 1675, Gottfried Leibniz was able to show the following Theorem:

Theorem: π/4 = 1 - 1/3 + 1/5 - 1/7 + ...

Proof:

(1) D(tan^-1x)= 1/(x² + 1) [See Theorem, here]

(2) So:

∫ dx/(x² + 1) = tan^-1 x [See Theorem 2, here]

(3) Since tan(0) = 0 and tan(π/4) = 1, we have:



(4) If we assume that (x² + 1) ≠ 0, then we have:

1/(x² + 1) = 1 - x² + x⁴ - x⁶ + x⁸ - ...

since:

(a) Let a = 1 - x² + x⁴ - x⁶ + x⁸ - ...

(b) Then:

x²*a = x² - x⁴ + x⁶ - x⁸ + ...

(c) Then:

a + x²*a = 1

(d) a + x²*a = a(1 + x²) = 1

(e) So:

a = 1/(1 + x²)

which means that:

1/(1 + x²) = 1 - x² + x⁴ - x⁶ + x⁸ - ...

(5) But this means that:



(6) Further, we know that:



1 - 1/3 + 1/5 - 1/7 + ...

(7) Putting step #3 together with step #5 and step #6 gives us:

π/4 = 1 - 1/3 + 1/5 - 1/7 + ...

QED

References

• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001