<s>[INST] What converges the series? $$\sum _{n=4}^{+\infty \:}\frac{4^n\sqrt{n+1}-4^n\sqrt{n}+2^{n-1}\sqrt{n}\sqrt{n+1}}{4^n\sqrt{n}\sqrt{n+1}}$$
I tried different ways to figure this out and at first i did integrals 
$$\int _4^{\infty \:}\frac{4^x\sqrt{x+1}-4^x\sqrt{x}+2^{x-1}\sqrt{x}\sqrt{x+1}}{4^x\sqrt{x}\sqrt{x+1}}dx$$
THEN 
$=0-\left(4-2\cdot \sqrt{5}-\frac{1}{32\ln \left(2\right)}\right)$
But then i realized it wasn't with integrals and that I can't do convergence D:
I just had to figure out what exactly converges the series without using integrals so I just exploded my brain with everything i did before.
 [/INST] $$\sum _{n=4}^{+\infty \:}\frac{4^n\sqrt{n+1}-4^n\sqrt{n}+2^{n-1}\sqrt{n}\sqrt{n+1}}{4^n\sqrt{n}\sqrt{n+1}} =\sum_{n=4}^{\infty} \left(\frac{1}{\sqrt{n}} -\frac{1}{\sqrt{n+1}}\right) +\sum_{n=4}^{\infty}\frac{1}{2^{n+1}} =\frac{1}{\sqrt{4}} +\frac{1}{16}$$
</s>