For the series 1/2 + 1 + 1/8 + 1/4 + 1/32 + 1/16 + 1/128 + 1/64 +... Does the series converge? Compute $\liminf (a_k)^{1/k}$ $\limsup (a_k)^{1/k}$ $\liminf (a_{k+1}/a_k)$ and $\limsup (a_{k+1}/a_k)$ as $k \rightarrow \infty$ .
Note. I think that the series can be rearrange in this way
(1 +1/2)+(1/8 + 1/4)+ (1/32 + 1/16) +..... = 3/2 ( 1 + 1/4 + 1/16 +.....)