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galactus · Feb 20, 2009, 05:44 PM

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

For example, he deduced that:

$\frac{1}{2}=1-1+1-1+.............$

and

$-1=1+2+4+8+.........$ by subbing in x=-1 and x=2 in the formula

$\frac{1}{1-x}=1+x+x^{2}+x^{3}+.......$

What was the error in his reasoning?

harum · Feb 20, 2009, 08:01 PM

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?