I wish to find a way to rearrange a series such that it diverges to infinity. If we take, for example, the alternating harmonic series: $$\sum_{n=1}^{\infty } \frac{(-1)^{n+1}}{n}$$
How can we rearrange this particular conditionally convergent series such that it diverges to $\infty$? Is there any simple trick for doing this? Also, if you could explain the process you used to find this rearrangement, that would be extremely helpful.
Thanks, Lauren.
Let $g(1)=1 .$ Let $a_{1,g(1)}=1 .$ Define $g(n+1)$, and $a_{j,g(n+1)}$ for $1\leq j \leq g(n+1)$ , recursively ,as follows : Given that $a_{n,g(n)}=1/(2 k_n-1)$ with $k_n\in Z^+ $, choose $g(n+1)$ large enough that $$\sum_{j=1}^{j=g(n+1)} (1/(2k_n+2j-1)>1 . $$ And let $a_{n+1,j}=1/(2k_n+2j-1)$ for $1\leq j\leq g(n+1) .$ Consider the series $$a_{1,g(1)}-1/2+a_{1,g(2)} +...+a_{g(2),g(2)}-1/4+a_{1,g(3)}+... +a_{g(3),g(3)} -1/6+a_{1,g(4)}+... $$ A conditionally convergent real series that is not absolutely convergent can actually be re-arranged into a series that sums to anything you want, including $\pm \infty$.