The series $$\frac{1}{2^2}+\frac{1}{2}+\frac{1}{2^4}+\frac{1}{2^3}+...$$ is a rearrangement of the geometric series $$\sum_{n=1}^\infty\frac{1}{2^n}$$
Does it converge or converge absolutely?
(Theorem: If the series $\sum_{n=1}^\infty a_n$ converges, then any series obtained by grouping the series of $\sum_{n=1}^\infty a_n$ is also convergent and has the same value as $\sum_{n=1}^\infty a_n$)
My confusion: Based on the theorem, the series $$\frac{1}{2^2}+\frac{1}{2}+\frac{1}{2^4}+\frac{1}{2^3}+...$$ is convergent since $\sum_{n=1}^\infty\frac{1}{2^n}$ is a geometric series with |r|<1, but if i work from the series $\frac{1}{2^2}+\frac{1}{2}+\frac{1}{2^4}+\frac{1}{2^3}+...$, my answer is not convergent. And furthermore, i found out that limsup of this series = 2 $\not\lt$ 1 i.e. the series do not converge absolutely.