# Some algebraic manipulation.

#### ensbana

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
Well it's bounded below by 1. I suspect it increases without bound as $k\to \infty$

• ensbana

#### idontknow

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

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

• ensbana

#### romsek

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

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

• ensbana, topsquark and idontknow

#### skipjack

Forum Staff
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

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.

• idontknow

#### idontknow

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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