I know the definitions and tests for convergence, my question deals with an alternating series which converge on a range. For example, we can consider the sum of the alternating harmonic series: $$ \sum_{n = 1}^{\infty} \dfrac{(-1)^{n+1}}{n} =1-\dfrac{1}{2}+\dfrac{1}{3} - \dfrac{1}{4} + \dfrac{1}{5} \dots =\ln2 $$ Now, suppose we were to modify this series, into one which does not converge on a single value, but alternates between two values, to both of which it converges on each sign change. For example, We change the above mentioned series in the following way:
$$ \sum_{n = 1}^{\infty} \dfrac{(n+1)(-1)^{n+1}}{n} =\dfrac{2}{1}-\dfrac{3}{2}+\dfrac{4}{3} - \dfrac{5}{4} \dots $$
This series does not converge according to the traditional definition of convergence I guess, failure to converge on a single value is considered divergence- and I know that, but it does alternate between two values on which it seems to converge, those values seem to be (just by looking at the partial sums): $$ \sum_{n = 1}^{\infty} \dfrac{(n+1)(-1)^{n+1}}{n} = \begin{cases} \ln2 &\quad \text{if} \quad n \mod2 \equiv 0 \\ \ln2 + 1 &\quad \text{if} \quad n \mod2 \equiv 1\\ \end{cases} $$
My question is, is there some framework to study these kinds of series? Even though the series does not converge on a point, it is still kind of cool if we might be able to say that both of its branches are convergent with a definite value between them