Thread: difference of a diverging series

1. difference of a diverging series

Hello,
If $\sum_{1}^{\infty} a_n$ is a diverging series, is $\sum_{1}^{\infty} a_n - \sum_{1}^{\infty} a_n$ equal to zero, or is it undefined?

2. Since they are exactly the same in this case it would equal 0.

If however, it was the difference of two different diverging series, it would be indeterminate.

3. Originally Posted by dudyu
Hello,
If $\sum_{1}^{\infty} a_n$ is a diverging series, is $\sum_{1}^{\infty} a_n - \sum_{1}^{\infty} a_n$ equal to zero, or is it undefined?
That is a case where You must be carefull!... if $\displaystyle \sum_{n=1}^{\infty} a_{n}$ diverges, the following expressions are not the same...

a) $\displaystyle \sum_{n=1}^{\infty} a_{n} - \displaystyle \sum_{n=1}^{\infty} a_{n}$

b) $\displaystyle \sum_{n=1}^{\infty} \{a_{n} - a_{n}\}$

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

$\chi$ $\sigma$