Intereting Posts

How to find an ellipse, given five points?
How to prove that $\sum _{k=0}^{2n-1} \frac{(-2n)^k}{k!}<0 $
Matrix with zeros on diagonal and ones in other places is invertible
Mathematical trivia (i.e. collections of anecdotes and miscellaneous (recreational) mathematics)
Is the notion of a set always a primitive notion?
Which are the Bound and Free Variables in these expressions?
Is B(H) a Hilbert space?
A connected k-regular bipartite graph is 2-connected.
Variance decomposition over pairs of elements
Compendium(s) of Elementary Mathematical Truths
Prove that $|\log(1 + x^2) – \log(1 + y^2)| \le |x-y|$
Is there any geometric way to characterize $e$?
Equivalent metrics determine the same topology
Show that $\sqrt{4+2\sqrt{3}}-\sqrt{3}$ is rational using the rational zeros theorem
Convergence of $\frac{\sqrt{a_{n}}}{n}$

show that $$\sum\limits_{i=1}^n \frac{x_i}{i^2} \geq \frac{1}{1} + \frac{1}{2} + \dots +\frac{1}{n}$$

where $x_1,x_2,\dots,x_n$ are natural numbers and all of them are different numbers(no such a $x_i=x_j$)

the teacher said you can prove it by making it a Cauchy form inequality.

thing i have tried to make Cauchy inequality and show it’s same as question inequality:

multiply left side by $(1^2+2^2…+n^2)$.

- Induction with floor, ceiling $n\le2^k\implies a_n\le3\cdot k2^k+4\cdot2^k-1$ for $a_n=a_{\lfloor\frac{n}2\rfloor}+a_{\lceil\frac{n}2\rceil}+3n+1$
- Inequality $\sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18}$
- Proof that $\sum\limits_{j,k=1}^N\frac{a_ja_k}{j+k}\ge0$
- Show that $\, 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 $
- Why does the sign have to be flipped in this inequality?
- Given that a,b,c are distinct positive real numbers, prove that (a + b +c)( 1/a + 1/b + 1/c)>9

multiply right side by $$\sum\limits_{i=1}^n \frac{i^2}{x_i}$$

and in none of them i was successful.

- $L^2$ norm inequality
- An Inequality Involving Bell Numbers: $B_n^2 \leq B_{n-1}B_{n+1}$
- $|f(x)-f(y)|\le(x-y)^2$ without gaplessness
- System of Equations: any solutions at all?
- Prove that $\sum_{k=1}^n \frac{2k+1}{a_1+a_2+…+a_k}<4\sum_{k=1}^n\frac1{a_k}.$
- How prove this inequality $(a^2+bc^4)(b^2+ca^4)(c^2+ab^4) \leq 64$
- Proof of upper-tail inequality for standard normal distribution
- Complex inequality $||u|^{p-1}u - |v|^{p-1}v|\leq c_p |u-v|(|u|^{p-1}+|v|^{p-1})$
- If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $(a+1/a)^2+(b+1/b)^2\ge 25/2$
- Can this inequality proof be demystified?

A proof just with Cauchy-Schwarz:

From Cauchy-Schwarz, one have that

$$

(\sum_i \frac{1}{x_i})(\sum_i \frac{x_i}{i^2}) \ge (\sum_i \frac{1}{i})^2.

$$

But since $\sum_i \frac{1}{x_i} \le \sum_i \frac{1}{i}$, one have that

$$

\sum_i \frac{x_i}{i^2} \ge

\frac{(\sum_i \frac{1}{i})^2}{\sum_i \frac{1}{x_i}} \ge

\frac{\sum_i \frac{1}{i}}{\sum_i \frac{1}{x_i}} \sum_i \frac{1}{i} \ge

\sum_i\frac{1}{i}.

$$

By the Rearrangement Inequality (but we don’t need anything that general) the left side is minimized, for fixed $x_i$, if the $x_i$ are increasing. And then the minimum is reached if the $x_i$ are as small as possible, which gives $x_i=i$.

**Remark:** If we don’t want to quote the Rearrangement Inequality, it is clear that if $i\lt j$ and $x_i \gt x_j$, then $\frac{x_i}{i^2}+\frac{x_j}{j^2} \gt \frac{x_j}{i^2}+\frac{x_i}{j^2}$.

To give – in Addition to Andres answer – a solution using the Cauchy-Schwarz inqquality, we let $\xi_i := \frac{\sqrt{x_i}}i$, and $\eta_i := \frac{1}{\sqrt{x_i}}$. Then, by Cauchy-Schwarz

\begin{align*}

\sum_i \frac 1i &= \sum_{i} \xi_i \eta_i \\ &\le \left(\sum_{i} \xi_i^2\right)^{1/2} \left(\sum_i \eta_i^2\right)^{1/2}\\

&= \left(\sum_i \frac{x_i}{i^2}\right)^{1/2} \left(\sum_i \frac 1{x_i}\right)^{1/2}

\end{align*}

Now choose a $\sigma \in S_n$ such that $x_{\sigma(1)} < \ldots < x_{\sigma(n)}$, then $x_{\sigma(i)}\ge i$, giving

$$

\sum_i \frac 1{x_i} = \sum_j \frac 1{x_{\sigma(j)} }\le \sum_j \frac 1j

$$

Continuing above

$$ \sum_i \frac 1i \le \left(\sum_i \frac{x_i}{i^2}\right)^{1/2} \left(\sum_i \frac 1i\right)^{1/2} $$

Hence diving by $\left(\sum \frac 1i\right)^{1/2}$ and squaring gives the result.

- What is wrong in my proof that 90 = 95? Or is it correct?
- Petersen graph is not a Cayley graph
- Does every strongly convex function has a stationary point?
- How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?
- Hilbert's Original Proof of the Nullstellensatz
- $S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$: Show uniformly converges within $$
- Standard example where Jacobson radical not equal nilradical
- Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$ is a noninteger
- True or False $A – C = B – C $ if and only if $A \cup C = B \cup C$
- A closed form for $\int_0^\infty\ln x\cdot\ln\left(1+\frac1{2\cosh x}\right)dx$
- About the determination of complex logarithm
- Is Fourier series an “inverse” of Taylor series?
- Example of a non-Noetherian complete local ring
- Complex roots of $z^3 + \bar{z} = 0$
- If any x points are elected out of a unit square, then some two of them are no farther than how many units apart?