Sequence Epsilon

[Toymen]

Show that the sequence [MATH] cn = \sqrt{7n + 19} - \sqrt{4n} [/MATH] is certainly divergent against [MATH]oo[/MATH]

This method should return a natural number [MATH]n_2[/MATH] for [MATH]M >0 [/MATH],
so that [MATH]∀n ≥ n2 : cn ≥ M[/MATH]
A formula for the calculation of [MATH]n2 [/MATH] results from your proof.

 

[LCKurtz]

Hint: Multiply it by [MATH]\frac{\sqrt{7n+19}+\sqrt{4n}}{\sqrt{7n+19}+\sqrt{4n}}[/MATH] and simplify to get [MATH]\frac{3n+19}{\sqrt{7n+19}+\sqrt{4n}}[/MATH]. Then underestimate the numerator and overestimate the denominator to get something smaller and simpler that still goes to infinity.