Determine if the following series converges or diverges and justify your decision.
![\[ \sum_{n=1}^{\infty} \left( \frac{1}{n} - e^{-n^2} \right). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-4b0fd040079b22808f4c8d4e8479ffba_l3.png "Rendered by QuickLaTeX.com")
We know from the previous exercise that the series
![\[ \sum_{n=1}^{\infty} e^{-n^2} \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-f5a16e85f691d5bb02267352fb37d130_l3.png "Rendered by QuickLaTeX.com")
converges. Hence, the series in this exercise must diverge. Otherwise, the convergence of the series in the exercise and the convergence of 
would imply that 
converges, which is false. Hence, the series
![\[ \sum_{n=1}^{\infty} \left( \frac{1}{n} - e^{-n^2} \right) \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-7450dfbe647ee32696ee5b481cb8d03f_l3.png "Rendered by QuickLaTeX.com")
must diverge.