I am aware that one is unable to rearrange terms in a conditionally convergent series. But, take a conditionally convergent series, say
$$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$$
and group terms with a stride of two to produce
$$\sum_{n=1}^{\infty} \frac{1}{2n-1} – \frac{1}{2n}.$$
With this particular series, even if I take the absolute value of the terms, it converges to the same value. Does this imply the reformatted series is absolutely convergent, and that I can validly rearrange its terms arbitrarily?
Best Answer
Yes, the new series you have created is absolutely convergent, and rearranging its terms will not change its value. Note however, that the set of terms of the new series, $$ b_n = \frac{1}{2n-1} - \frac{1}{2n} = \frac{1}{2n(2n-1)}, $$ is not the same as the set of terms of the old series, so this is not a rearrangement of the old series in the strict sense.