Consider the functional sequence \begin{align*} f_n(x) = \sqrt{n+1} - \sqrt{n+2} + \sum_{k=1}^n \frac{1}{2\sqrt{k+x+2}} - \frac{1}{2\sqrt{k+x+1}}, (1\leq x\leq 2, n=1,2,3,...). \end{align*} Is $f_n(x)$ uniformly convergent on $[1, 2]$?
I know that $f(x) = \displaystyle{\lim_{n\to\infty}}f_n(x) = \sum_{k=1}^\infty\frac{1}{2\sqrt{k+x+2}} - \frac{1}{2\sqrt{k+x+1}} = \frac{-1}{2\sqrt{x+2}}$, but I do not know how to use the supremm test. Please guide me? (Let $M_n = sup|f_n(x) - f(x)|$, so $f_n\rightarrow f$ is uniformly convergent on $E$ if and only if $M_n\rightarrow 0$ as $n\rightarrow \infty$).
For $x\in[1,2]$, $$ \begin{align} f_n(x) &=\sqrt{n+1}-\sqrt{n+2}+\sum_{k=1}^n\left(\frac1{2\sqrt{k+x+2}}-\frac1{2\sqrt{k+x+1}}\right)\\ &=-\underbrace{\frac1{\sqrt{n+1}+\sqrt{n+2}}}_{\le\frac1{2\sqrt{n+1}}}+\underbrace{\frac1{2\sqrt{x+n+2}}}_{\le\frac1{2\sqrt{n+3}}}-\frac1{2\sqrt{x+2}}\\ \end{align} $$ Can you finish from here?