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

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.

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.