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Proof of Euler formula

The expansion

tan^(-1) x = x - x^3/3 + x^5/5 - x^7/7 + ...

at typeset structure cannot be used, since the series expansion is valid only for typeset structure. Instead start with the finite form of the geometric series expansion

1/(1 + t^2) = 1 - t^2 + t^4 + ... + (-1)^n t^(2 n) + (-1)^(n + 1) t^(2 n + 2)/(1 + t^2) .

Integrating from 0 to 1 gives

π/4 = tan^(-1) 1 = 1 - 1/3 + 1/5 - ... + (-1)^n/(n + 1) + (-1)^(n + 1) ∫ t^(2 n + 2)/(1 + t^2) d t .

We are done, since the last term tends to zero as typeset structure

0 <= ∫ t^(2 n + 2)/(1 + t^2) d t <= ∫ t^(2 n + 2) d t = 1/(2 n + 3) .

Cite this web-page as:

Štefan Porubský: Proof of Euler formula.

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