Leonhard Euler made occasional errors in his reasoning regarding infinite series.

For example, he deduced that:



and

by subbing in x=-1 and x=2 in the formula



What was the error in his reasoning?

He probably didn't do his homework. The formula is a Maclaurin series expansion for 1/(1-x), which is always around x=0. *x=0* If you, or he, want to have series expansion for 1/(1-x) around a random point x0, then more general Taylor series and not Maclaurin series should be used: 1/(1-x)= 1/(1-x0) + (x-x0)/(1-x0)^2 + (x-x0)^2/(1-x0)^3 +... In case he considered this formula as the sum of infinitely decreasing geometric sequence, then it does not work in this particular case as x has to be less than 1 and more than -1. Do you have a reference?