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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 $k \rightarrow \infty$

For the series $$\frac{1}{2} + 1 + \frac{1}{8} + \frac{1}{4} + \frac{1}{32} + \frac{1}{16} + \frac{1}{128} + \frac{1}{64} +...$$ Compute

  1. $\liminf_{k\to\infty}(a_k)^{1/k}$

  2. $\limsup_{k\to\infty}(a_k)^{1/k}$

  3. $\liminf_{k\to\infty}(a_{k+1}/a_k)$ and

  4. $\limsup_{k\to\infty}(a_{k+1}/a_k)$.

This is geometric series. But this does not order. So how I can find?

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    Why would you ask the same question twice when it has been answered? If my answer to that question did not suffice, then you could tell me in the comments to add more details!2012-12-28
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    Ok. I cannot understand. Please explain more and more clearly. Please. I need to know how to solve those precisely. I need! If you mind, Can you solve clearly? Thank you so much @nameless2012-12-28
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    Ok thank you so much:)) @nameless2012-12-28
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    Ok @nameless I wait your solution:)2012-12-28
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    @nameless sorry I'm elementary math student. I start learn new.2012-12-28
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    You may be able to see that the terms it even positions follow a simple pattern, as do the terms in the odd positions. For the first two it may be simpler to look at odd and even terms separately.2012-12-28

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The ratios $\dfrac{a_{k+1}}{a_k}$ are easiest to deal with. Note that they are alternately $2$ and $\dfrac{1}{8}$. Now the $\liminf$ and $\limsup$ should be easy to write down.

For the roots, call the terms of our sequence $a_1,a_2,a_3,a_4,\dots$.

Compute $a_k^{1/k}$ for the first few terms. It will tell you what's going on.

But we are a little impatient, so will go to formulas. Note that our sequence goes $$2^{-1}, 2^{-0}, 2^{-3}, 2^{-2}, 2^{-5}, 2^{-4}, 2^{-7}, 2^{-6},\dots.$$

So if $k$ is odd, then $a_k=2^{-k}$. If $k$ is even, then $a_k=2^{-(k-2)}=\dfrac{2^{-k}}{4}$.

Now take $k$-th roots. If $k$ is odd, then $a_k^{1/k}=2^{-1}$.

If $k$ is even, then $a_k^{1/k}=\dfrac{2^{-1}}{4^{1/k}}$. Now it should not be hard to answer the questions about $\liminf$ and $\limsup$.

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    Thank you very much. So clear solution. Thank you2012-12-28
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    Note that there is some work left for you to do, well, not really in the ratio case!2012-12-28