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 [MATH]k[/MATH] a positive integer, and [MATH]\delta \in (0,1)[/MATH]: [MATH]\frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta}[/MATH]

 

[pka]

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 [MATH]k[/MATH] a positive integer, and [MATH]\delta \in (0,1)[/MATH]: [MATH]\frac{\sqrt{(1- \delta)^{2k} + 4\delta} \ - (1-\delta)^k}{2\delta}[/math]

Multiply \(\displaystyle \frac{{\sqrt {{{(1 - \delta )}^{2k}} + {{(1 - \delta )}^k}} }}{{2\delta }}\left( {\frac{{\sqrt {{{(1 - \delta )}^{2k}} - {{(1 - \delta )}^k}} }}{{\sqrt {{{(1 - \delta )}^{2k}} - {{(1 - \delta )}^k}} }}} \right)\)