Showing that the inequality $\frac{\sqrt{2}}{n+1} - \frac{1}{2n^2} \leq u_n \leq \frac{\sqrt{2}}{n+1}$ stands.

Question

If $\displaystyle u_{n}=\int_{0}^{1} t^n \sqrt{t+1} dt$, where $n\geq 1$, prove that $$\frac{\sqrt{2}}{n+1} - \frac{1}{2n^2} \leq u_n \leq \frac{\sqrt{2}}{n+1}.$$

I have already proven the second inequality, but couldn't the first one. I managed to show that $\dfrac{1}{n+1} \leq u_n$, and I wrote a recursive formula by using the $\displaystyle \int_{0}^{1} \dfrac{t^n}{\sqrt{t+1}} dt$ integral, it got worse. Seems to be some easier way.

Answer 1

Since $t\to\sqrt{t+1}$ is concave, it follows that, for $t\in [0,1]$, $$1+(\sqrt{2}-1)t\leq \sqrt{t+1}\leq \sqrt{2}.$$ Hence $$\frac{1}{n+1}+\frac{\sqrt{2}-1}{n+2}\leq \int_{0}^{1} t^n \sqrt{t+1} dt\leq \frac{\sqrt{2}}{n+1}.$$ It remains to show that $$ \frac{\sqrt{2}}{n+1} - \frac{1}{2n^2}\leq \frac{1}{n+1}+\frac{\sqrt{2}-1}{n+2}$$ that is $$\frac{\sqrt{2}-1}{(n+1)(n+2)}\leq \frac{1/2}{n^2}$$ which holds for all $n\geq 1$.

Answer 2

$$\int_{0}^{1} t^n \sqrt{t+1} dt\leq\int_{0}^{1} t^n \sqrt{1+1} dt=\frac{\sqrt{2}}{n+1}$$ For the left hand side, with Beta function \begin{align} \int_{0}^{1} t^n (\sqrt{2}-\sqrt{t+1}) dt &= \int_{0}^{1} t^n \dfrac{1-t}{\sqrt{2}+\sqrt{t+1}}\ dt \\ &\leq \dfrac12\int_{0}^{1} t^n(1-t)\ dt \\ &= \frac12\beta(n+1,2) \\ &= \dfrac{1}{2(n+1)(n+2)}\leq\dfrac{1}{2n^2} \end{align}

Answer 3

$\sqrt{t+1}$ is approximately constant on $(0,1)$, hence the following application of Cauchy-Schwarz

$$ I_n=\int_{0}^{1}t^n\sqrt{1+t}\,dt\leq \sqrt{\int_{0}^{1}t^{n}\,dt\int_{0}^{1}t^n(1+t)\,dt}=\color{red}{\frac{1}{n+1}\sqrt{\frac{2n+3}{n+2}}}<\frac{\sqrt{2}}{n+1}$$ is expected to provide a tight upper bound. Indeed, $$ \frac{\sqrt{2}}{n+1}-I_n = \int_{0}^{1}t^n\left(\sqrt{2}-\sqrt{1+t}\right)\,dt = \int_{0}^{1}(1-t)^n\left(\sqrt{2}-\sqrt{2-t}\right)\,dt $$ leads to $$\begin{eqnarray*}\frac{\sqrt{2}}{n+1}-I_n = \int_{0}^{1}\frac{t(1-t)^n}{\sqrt{2}+\sqrt{2-t}}\,dt&\leq&\frac{1}{\sqrt{2}+1}\int_{0}^{1}t(1-t)^n\,dt\\&=&\frac{1}{\sqrt{2}+1}\cdot\frac{1}{(n+1)(n+2)}\\&<&\frac{1}{2(n+1)^2}.\end{eqnarray*}$$ Actually we have $$ I_n = \frac{\sqrt{2}}{n+1}+\frac{C_n}{(n+1)^2} $$ where $C_n\in\left[\frac{1}{2\sqrt{2}},\frac{1}{\sqrt{2}+1}\right]$.