∫-0.50.5 1/(1-x) dx
= ∫-0.50.5 1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - ..... dx |Summandenweise integrieren (nachdem bewiesen ist, dass man das darf)
= x - 1/2 x^2 + 1/3 x^3 - 1/4 x^4 + 1/5 x^5 - ..... |-1/21/2
= (1/2 - (-1/2)) - 1/2 ( 1/2^2 - 1/2^2) + 1/3 ((1/2)^3 - (-1/2)^3) - 1/4 ( ......) + ....
= 1 - 0 + 1/3 (2*(1/2)^3) -0 + 1/5 (2*(1/2)^5) - 0 + ....
= 1 - 0 + 1/3 ((1/2)^2) -0 + 1/5 ((1/2)^4) - 0 + 1/7 (1/2)^6 +1/9 (1/2)^8
= Σ_(k=0)^{∞} 1/(2k+1) * (1/2)^{2k}
= Σ_(k=0)^{∞} 1/(2k+1) * (1/4)^{k}