Let $n\in \mathbb N$ and:
$$
x_n = \sqrt{\sum_{k=1}^n\left(\frac{k+1}{k}\right)^2} – \sqrt n
$$
Show that $\{x_n\}$ is a bounded sequence.
This sequence appears a bit tricky because it involves harmonic series. Below are steps I take.
Lower bound:
$$
\sqrt{\sum_{k=1}^n\left(\frac{k+1}{k}\right)^2} \ge \sqrt{\sum_{k=1}^n\left(\frac{k}{k}\right)^2} = \sqrt{\sum_{k=1}^n1}=\sqrt{n}\implies\\
\implies x_n \ge \sqrt n – \sqrt n \ge 0
$$
Lower bound is simple.
Upper bound:
To get rid of radical lets use Cauchy-Schwarz (note the below is incorrect as shown in Zvi's answer):
$$
\sqrt{\sum_{k=1}^n\left(\frac{k+1}{k}\right)^2} \le \sqrt{\left(\sum_{k=1}^n\left(\frac{k+1}{k}\right)\right)^2} =\sum_{k=1}^n\left(\frac{k+1}{k}\right) = n+\sum_{k=1}^n{1\over k} = n + H_n
$$
So this doesn't show $x_n$ is bounded above. I've tried another approach:
$$
\sqrt{\sum_{k=1}^n\left(\frac{k+1}{k}\right)^2} – \sqrt n = \frac{\sum_{k=1}^n\left(\frac{k+1}{k}\right)^2 – n}{\sqrt{\sum_{k=1}^n\left(\frac{k+1}{k}\right)^2} + \sqrt n}
$$
Consider nominator:
$$
\sum_{k=1}^n\left(\frac{k+1}{k}\right)^2 – n=n+\sum_{k=1}^n{2\over k}+\sum_{k=1}^n{1\over k^2}-n = \sum_{k=1}^n{1\over k^2} + \sum_{k=1}^n{2\over k}
$$
For denominator:
$$
\sqrt{\sum_{k=1}^n\left(\frac{k+1}{k}\right)^2} + \sqrt n = \sqrt{n + \sum_{k=1}^n{1\over k^2}+\sum_{k=1}^n{2\over k}} + \sqrt n
$$
So $x_n$ is:
$$
x_n = \frac{\sum_{k=1}^n{1\over k^2} + \sum_{k=1}^n{2\over k}}{ \sqrt{n + \sum_{k=1}^n{1\over k^2}+\sum_{k=1}^n{2\over k}} + \sqrt n }
$$
But i don't see how to proceed from this point. What else could i try? How to show $x_n$ is bounded above?
Please note the precalculus tag.
Best Answer
Jakobian gave a very good hint. I am giving somewhat sharper bounds for $x_n$. Note that the OP incorrectly used the Cauchy–Bunyakovsky–Schwarz inequality. Indeed, when applied correctly, we get $$\sqrt{n}\left(x_n+\sqrt{n}\right)=\sqrt{\sum_{k=1}^n1^2}\sqrt{\sum_{k=1}^n\left(\frac{k+1}{k}\right)^2}\geq \sum_{k=1}^n1\cdot\left(\frac{k+1}{k}\right)=n+H_n,$$ where $H_n=\sum_{k=1}^n\frac1k$ is the $n$th harmonic number. That is, $$x_n\geq \frac{H_n}{\sqrt{n}}.$$
For the upper bound, the OP got $$x_n=\frac{\sum_{k=1}^n \frac{1}{k^2}+2H_n}{\sqrt{\sum_{k=1}^n\left(\frac{k+1}{k}\right)^2}+\sqrt{n}}.$$ We can show that $\sum_{k=1}^n\frac{1}{k^2}$ is bounded above by $2$ via $$\sum_{k=1}^n\frac{1}{k^2}<1+\sum_{k=2}^n\frac{1}{k(k-1)}=1+\left(1-\frac{1}{n}\right)<2.$$ However, a sharper upper bound is $\sum_{k=1}^n\frac{1}{k^2}<\zeta(2)=\frac{\pi^2}{6}$. Also, $$\sum_{k=1}^n\left(\frac{k+1}{k}\right)^2\geq \sum_{k=1}^n1^2=n.$$ This proves that $$x_n<\frac{\frac{\pi^2}{6}+2H_n}{\sqrt{n}+\sqrt{n}}=\frac{H_n}{\sqrt{n}}+\frac{\pi^2}{6}\left(\frac{1}{2\sqrt{n}}\right).$$ (You can replace $\frac{\pi^2}{6}$ by $2$ if you want a precalculus solution.)
That is, we have $$\frac{H_n}{\sqrt{n}}\leq x_n<\frac{H_n}{\sqrt{n}}+\frac{\pi^2}{6}\left(\frac{1}{2\sqrt{n}}\right).$$ So, $x_n$ has the asymptotic behavior of $\frac{H_n}{\sqrt{n}}\approx \frac{\ln n}{\sqrt{n}}$. But to show that $x_n$ is bounded, we don't need to know that $H_n\approx \ln n$ (well, to be even more precise, $H_n\approx \gamma+\ln n$, where $\gamma$ is the Euler–Mascheroni constant). Clearly, $$\frac{\pi^2}{6}\left(\frac{1}{2\sqrt{n}}\right)\leq \frac{\pi^2}{6}\left(\frac{1}{2}\right).$$ Furthermore, $$H_n=\sum_{k=1}^n\frac{1}{k}\leq \sum_{k=1}^n\frac{1}{\sqrt{k}}<\sum_{k=1}^n\frac{2}{\sqrt{k}+\sqrt{k-1}}=2\sum_{k=1}^n(\sqrt{k}-\sqrt{k-1})=2\sqrt{n}.$$ Hence, $$x_n<\frac{2\sqrt{n}}{\sqrt{n}}+\frac{\pi^2}{6}\left(\frac{1}{2}\right)=2+\frac{\pi^2}{6}\left(\frac{1}{2}\right)<\infty.$$ Again, replace $\frac{\pi^2}{6}$ by $2$ if you don't want any non-precalculus knowledge in the proof. So, you would get $x_n<3$ for all $n$.