I want to show that
$$ \left(1-\frac{1}{2^r} \right) + \left( \frac{1}{3^r} - \frac{1}{4^r} \right) + \left( \frac{1}{5^r} - \frac{1}{6^r} \right) + \cdots $$ is absolutely convergent, where $0 < r \le 1$.
My idea is to use the comparison test and find a convergent series that is greater than or equal to the series above. I am not sure which series to use though.
Your series is $$\sum \left( \frac{1}{(2k-1)^r}- \frac{1}{(2k)^r}\right)$$
For all $k \geq 1$, you have $$ \frac{1}{(2k-1)^r}- \frac{1}{(2k)^r} \geq 0$$
So the absolute convergence of your series in the same as its convergence, which can be proved by the general criteria for alternating series.