# Absolute convergence

The answer is that because S is not absolutely convergent, rearranging its terms changes the value of the sum. This means {estilo de exibição S_{1}} e {estilo de exibição S_{2}} are not equal. Na verdade, the series {estilo de exibição 1-1+1-1+...} does not converge, so S does not have a value to find in the first place. A series that is absolutely convergent does not have this problem: rearranging its terms does not change the value of the sum.

Explanation This is an example of a mathematical sleight of hand. If the terms of S are rearranged in such a way that every term remains in its original position, one finds that S is either the infinite series {displaystyle S=1-1+1-1+...+1-1+1-1} or with equal possibility, este {displaystyle S=1-1+1-1+...+1-1+1} Evaluating S as before, by grouping every -1 with the +1 preceding it or by grouping every +1 except the first with the -1 preceding it, gives in the first case: {estilo de exibição S_{1}=(1-1)+....+(1-1)=0+....+0=0} {estilo de exibição S_{2}=1+(-1+1)+....+(-1+1)-1=1+0+....+0-1=1-1=0} e no segundo caso: {estilo de exibição S_{1}=(1-1)+....+(1-1)+1=0+....+0+1=1} {estilo de exibição S_{2}=1+(-1+1)+....+(-1+1)=1+0+...+0=1} This reveals the trick: the definition of S was interpreted as defining its last term as negative when evaluating {estilo de exibição S_{1}=0} but positive when evaluating {estilo de exibição S_{2}=1} when in fact the definition of S didn't define (and the rearrangement was independent of) either option.

Definition for real and complex numbers A sum of real numbers or complex numbers {textstyle sum _{n=0}^{infty }uma_{n}} is absolutely convergent if the sum of the absolute values of the terms {textstyle sum _{n=0}^{infty }|uma_{n}|} converge.