Find the minimum of $\sum_{k=1}^n \frac{x^k_k}{k}$

Let $n$ be a positive integer. Find the minimum of $\displaystyle \sum_{k=1}^n \dfrac{x^k_k}{k}$, where $x_1,x_2,\ldots,x_n$ are positive real numbers such that $\displaystyle \sum_{k=1}^n \dfrac{1}{x_k} = n$.

This question sort of reminds me of Cauchy-Schwarz but the fact that the denominators of the fractions in $\displaystyle \sum_{k=1}^n \dfrac{x^k_k}{k}$ are in increasing harmonic sequence, it seems hard to relate it. I can't really think of anything to do here.

• Simple Lagrange multiplier yields $x_k=C^{\frac 1{k+1}}$ where $C$ is determined from the constraint. – A.S. Jan 16 '16 at 4:24
• Is there a way without Lagrange multipliers? – Puzzled417 Jan 16 '16 at 4:27
• Lagrange multipliers are usually the way to go when optimizing with an equality constraint. – Ben Longo Jan 16 '16 at 4:35
• Minimum is achieved at $x_i=1$ ($C=1$). – A.S. Jan 16 '16 at 4:36

Lagrange multipliers are very straightforward for this problem: we get the equations $$x_j^{j-1}+\frac \lambda {x_j^2}=0,\ \ \ \ j=1,\ldots n.$$ Thus $x_1^2=x_2^3=\cdots =x_n ^{n+1}$, and so $$n=\sum_{k=1}^n\frac1 {x_k}=\sum_{k=1}^n \frac1 {x_1^{2/(k+1)}}.$$ If $x_1>1$, all terms in the sum are less than one and the equality is impossible; and if $x_1 <1$, all terms in the sum are greater than one, so again no equality. It follows that $x_1=\cdots=x_n=1$.
The constraint $$\sum_{k=1}^n\frac1{x_k}=n\tag{1}$$ implies that any infinitesimal change $\delta x_k$ satisfies $$\sum_{k=1}^n\frac{\delta x_k}{x_k^2}=0\tag{2}$$ To minimize $$\sum_{k=1}^n\frac{x_k^k}{k}\tag{3}$$ we need to find $x_k$ so that any $\delta x_k$ that satisfies $(2)$, gives $$\sum_{k=1}^nx_k^{k-1}\,\delta x_k=0\tag{4}$$ If $x_k^{k-1}$ is orthogonal to the same $\delta x_k$ that $\frac1{x_k^2}$ is, then there is a $\lambda$ so that $$x_k^{k-1}=\lambda\frac1{x_k^2}\tag{5}$$ which means $$x_k=\lambda^{\frac1{k+1}}\tag{6}$$ Combining $(1)$ and $(6)$ gives $$\sum_{k=1}^n\lambda^{-\frac1{k+1}}=n\tag{7}$$ Equation $(7)$ has the solution $\lambda=1$. Thus, the extreme value is \begin{align} \sum_{k=1}^n\frac{x_k^k}{k} &=\sum_{k=1}^n\frac{\lambda^{\frac{k}{k+1}}}{k}\\[3pt] &=H_n\tag{8} \end{align} where $H_n$ is the $n^{\text{th}}$ Harmonic Number.