,

where , , , …;

and ,

where , , , ….

For each series, Vega first calculated the coefficients , , , … and , , , …, then the terms of the series, and finally he summed the series. The smaller and larger parts were correct to 137 and 127 decimal places, respectively, so his calculation was correct up to 126 decimals. Had Vega done all the calculations correctly, he would have calculated to 140 decimals.

This Demonstration locates Vega's mistakes. The coefficients were all correct, but the terms were not. The crucial point was the calculation of , which was correct only to 127 decimals.