Hello,
If $\displaystyle \sum_{1}^{\infty} a_n$ is a diverging series, is $\displaystyle \sum_{1}^{\infty} a_n - \sum_{1}^{\infty} a_n $ equal to zero, or is it undefined?
That is a case where You must be carefull!... if $\displaystyle \displaystyle \sum_{n=1}^{\infty} a_{n}$ diverges, the following expressions are not the same...
a) $\displaystyle \displaystyle \sum_{n=1}^{\infty} a_{n} - \displaystyle \sum_{n=1}^{\infty} a_{n}$
b) $\displaystyle \displaystyle \sum_{n=1}^{\infty} \{a_{n} - a_{n}\}$
In a) each term of the 'sum' is a divergent series and the 'sum' is impossible. In b) You have a single series with all terms equal to 0, so that it converges and its sum is 0...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$