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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 +.....)

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Wait a minute here. This is not the geometric series but a rearrangement of that series. As you know, rearrangements of a series may diverge or even converge to different values than the original series.

You must show that this rearrangement converges to $2$ as well!

Hint: Absolute convergence implies all rearrangements of the series converge to the same value.

And why are all the limits $\frac12$?

EDIT: More details: Because the geometric series $\sum_{n=0}^{\infty}2^{-n}$ converges absolutely to $\frac{1}{1-\frac12}=2$ then so does this rearrangement. Your series theorefore converges to $2$.

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    I understand that part. But I cannot solve limsup liminf? Because I cannot use rearrangement of the series to find limsup liminf. I again use the given series in order to calculate limsup liminf. How?2012-12-28