I need to compute $$f(x)=\sum_{i=0}^\infty \left(\left\lfloor\frac{i}{2^x}\right\rfloor+x+1\right)(1-p)^{i-1}p$$and minimize it with respect to $x$ (an expression which will depend on $p$).
Compute infinite sum
1 Answers
You can break the sum up into pieces that can be handled as follows:
$$\frac{p}{1-p} \left [ (1+x) \sum_{i=0}^{\lceil{2^x}\rceil} (1-p)^i + (2+x) \sum_{i=\lceil{2^x}\rceil + 1}^{2 \lceil{2^{x}}\rceil} (1-p)^i + (3+x) \sum_{i=2 \lceil{ 2^{x}}\rceil + 1}^{3 \lceil{2^{x}}\rceil} (1-p)^i + \ldots \right ]$$
$$ =\frac{p}{1-p} \sum_{k=1}^{\infty} (k+x) \sum_{i=(k-1)\lceil{2^x}\rceil + 1}^{k \lceil{2^x}\rceil} (1-p)^i $$
You can see that this is going to end up as a simple geometric series (and its derivative):
$$ = \frac{p}{1-p} \sum_{k=1}^{\infty} (k+x) \left [ (1-p)^{(k-1)\lceil{2^x}\rceil} - (1-p)^{k\lceil{2^x}\rceil} \right ] $$
$$ = \frac{p^2}{1-p} \sum_{k=1}^{\infty} (k+x)\left[(1-p)^{\lceil{2^x}\rceil} \right ]^{k-1} $$
Using the fact that $\sum_{k=1}^{\infty} k q^{k-1}$ = 1/(1-q)^2, we get the following expression for the sum:
$$ = \frac{p^2}{1-p} \left [\frac{(1-p)^{\lceil{2^x}\rceil}}{1- (1-p)^{\lceil{2^x}\rceil}} + \frac{x}{\left [1- (1-p)^{\lceil{2^x}\rceil}\right ]^2} \right ] $$
As for minimizing the quantity represented by this expression, I would plot it for values of $x$ of interest.
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0Sorry, I can't :( – 2012-12-29
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0Use the fact that $\sum_{k=1}^{\infty} q^k = q/(1-q) $ and $\sum_{k=1}^{\infty} k q^k = q/(1-q)^2$. – 2012-12-29
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0Yes but I have k-1 and sum is from k – 2012-12-29
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0@Jessica, I'm not sure what you mean. – 2012-12-29
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0It would be great if you gave me more details on how it continues. – 2012-12-30
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0I am adding more steps. – 2012-12-30
$\sum_i(x+1)(1-p)^{i-1}p$
is finite for all $x$. – 2012-12-29