Some algebraic manipulation.

Feb 2020
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Germany
I've just got back to study some subjects that require maths, and my skill is pretty rusty. Could anyone help me with putting bounds on the following expression?

For \(\displaystyle k\) a positive integer, and \(\displaystyle \delta \in (0,1)\): \(\displaystyle \frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta}\)
 

romsek

Math Team
Sep 2015
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Well it's bounded below by 1. I suspect it increases without bound as $k\to \infty$
 
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Dec 2015
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Earth
$ \lim_{\delta \rightarrow 1 }
\frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta} =0$.

$
\lim_{\delta \rightarrow 0 }
\frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta} =\lim_{\delta \rightarrow 0 } \dfrac{4\delta }{4\delta }=1
$.
 
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romsek

Math Team
Sep 2015
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$\displaystyle\lim_{\delta \rightarrow 1 }
\frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta} =0$.

$
\displaystyle \lim_{\delta \rightarrow 0 }
\frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta} =\lim_{\delta \rightarrow 0 } \frac{4\delta }{2\delta }=2
$.
Mathematica returns a limit of 1 as $\delta \to 0$
 

skipjack

Forum Staff
Dec 2006
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The quoted post was corrected by changing the final denominator to $4\delta$, though changing the final numerator to $2\delta$ would have been more logical.
 

SDK

Sep 2016
801
543
USA
I've just got back to study some subjects that require maths, and my skill is pretty rusty. Could anyone help me with putting bounds on the following expression?

For \(\displaystyle k\) a positive integer, and \(\displaystyle \delta \in (0,1)\): \(\displaystyle \frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta}\)
Fix $\delta$ and apply the mean value inequality you can get the following estimate:
\[\frac{ \sqrt{(1 - \delta)^{2k} + 4 \delta} - \sqrt{(1 - \delta)^{2k}}}{4 \delta} \approx \frac{\partial_k (1 - \delta)^{2k}}{2 (1- \delta)^k} = \frac{\log(1- \delta)(1 - \delta)^{2k}}{(1 - \delta)^k.} = \log(1 - \delta)(1 - \delta)^k\]
This estimate differs from the one you want by multiplying by 2. This estimate can be made into an explicit upper/lower bounds by applying the MVT more precisely if necessary.
 
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Dec 2015
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for given domain \(\displaystyle (0,1)\) then :

\(\displaystyle \sup
\frac{ \sqrt{(1 - \delta)^{2k} + 4 \delta} - \sqrt{(1 - \delta)^{2k}}}{4 \delta} =\sup
\log(1 - \delta)(1 - \delta)^k


\) occurs for smaller \(\displaystyle \delta\) , which is \(\displaystyle \delta \rightarrow 0\).

\(\displaystyle \inf
\frac{ \sqrt{(1 - \delta)^{2k} + 4 \delta} - \sqrt{(1 - \delta)^{2k}}}{4 \delta}
=\inf
\log(1 - \delta)(1 - \delta)^k

\) occurs for larger \(\displaystyle \delta \) , which is \(\displaystyle \delta \rightarrow 1\).
 
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