Intereting Posts

Variance of time to find first duplicate
Number of all labeled, unordered rooted trees with $n$ vertices and $k$ leaves.
Easy way to find roots of the form $qi$ of a polynomial
Sanity check about Wikipedia definition of differentiable manifold as a locally ringed space
Does $IJ=IK\implies J=K$ always hold for integral domain and finitely generated nonzero ideal $I$?
Book request: Mathematical Finance, Stochastic PDEs
Maximum principle – bounds on solution to heat equation with complicated b.c.'s
Evaluating the integral $\int_0^{\frac{\pi}{2}}\log\left(\frac{1+a\cos(x)}{1-a\cos(x)}\right)\frac{1}{\cos(x)}dx$
Jech's proof of Silver's Theorem on SCH
Galois Group of $x^{4}+7$
Degree of the extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$
Defining the Initial Conditions for a Planetary Motion to Have a Circular Orbit.
what is the cardinality of set of all smooth functions in $L^1$?
In a integral domain every prime element is irreducible
Integer solutions of a cubic equation

Looking at an algorithm for minimizing $\sum_{k=1}^{m} \frac{1}{n_k}\ln n_k > 1$ subject to $\sum_{k=1}^{m}\frac{1}{n_k} = 1$ in which $n_k$ are positive and in general non-sequential integers, I wondered about the more general problem of finding the minimum of $\sum_{k=1}^{m} \frac{1}{n_k}\ln n_k > 1$ subject to $\sum_{k=1}^{m}\frac{1}{n_k} \simeq 1$.

For example: $\frac{1}{2}+ \frac{1}{3}+\frac{1}{6} = 1$, and $\frac{1}{2}\ln 2+ \frac{1}{3}\ln 3 + \frac{1}{6}\ln 6 \simeq 1.014$.

We also have $\frac{1}{2}+\frac{1}{3}+\frac{1}{8} + \frac{1}{200}+\frac{1}{5000} \simeq .96 $ with

- Help with an integral for: If U has a $\chi^2$ distribution with v df, find E(U) and V(U)
- Prove that $\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots.$
- Integral ${\large\int}_0^1\frac{\ln^2\ln\left(\frac1x\right)}{1+x+x^2}dx$
- How to solve $\int_0^\pi{\frac{\cos{nx}}{5 + 4\cos{x}}}dx$?
- Evaluate $\int_0^\infty\frac{1-e^{-x}(1+x )}{x(e^{x}-1)(e^{x}+e^{-x})}dx$
- Estimate a sum of products

$\frac{1}{2}\ln 2 +\frac{1}{3}\ln 3 + \frac{1}{8}\ln 8 + \frac{1}{200}\ln 200 +\frac{1}{5000}\ln 5000 \simeq 1.0009$.

Are there general ways of thinking about this? While I would think there are a finite number of solutions for $(\sum_{k=1}^{m} \frac{1}{n_k}\ln n_k – 1 )< \epsilon_1$ and $| \sum_{k=1}^{m}\frac{1}{n_k} – 1| \leq \epsilon_2$, and a countable number of solutions if m can be infinite, I don’t see any systematic way of finding solutions even in the finite case.

Thanks for any suggestions.

Edit: typo corrected–sense of inequality in $\epsilon_1$ expression was backward. Should conform to question in title and first paragraph above.

- Can a function with just one point in its domain be continuous?
- proof of l'Hôpital's rule
- Today a student asked me $\int \ln (\sin x) \, dx.$
- Why does factoring eliminate a hole in the limit?
- Taking the half-derivative of $e^x$
- How to find points of tangency on an ellipse?
- calculation of $\int\frac{1}{\sin^3 x+\cos^3 x}dx$ and $\int\frac{1}{\sin^5 x+\cos^5x}dx$
- Show that, for all $n > 1: \log \frac{2n + 1}{n} < \frac1n + \frac{1}{n + 1} + \cdots + \frac{1}{2n} < \log \frac{2n}{n - 1}$
- Calculate Inverse Laplace transform
- How to prove $\sin(1/x)$ is not uniformly continuous

[EDITED: This answer isn’t relevant to the updated version of the question.] Given any (large) integer $B$, if you let $n_1=B$, …, $n_m = [eB]$, then

$$

\sum_{k=1}^m \frac1{n_k} = 1 + O(1/B)

$$

but

$$

\sum_{k=1}^m \frac{\ln n_k}{n_k} > \ln B \sum_{k=1}^m \frac1{n_k} > \ln B + O(1).

$$

So there are lots of solutions where the sum with ln is far larger than 1.

- Solving $\cos x+\sin x-1=0$
- How to prove that $\omega (n) = O\Big{(} \frac{\log(n)}{\log(\log(n))}\Big{)}$ as $n \to \infty$?
- Showing uniform continuity
- Family of geometric shapes closed under division
- Are Clifford algebras and differential forms equivalent frameworks for differential geometry?
- A map which commutes with Hodge dual is conformal?
- Proving $x^{\log n} = n^{\log x}$
- Chain rule for matrix exponentials
- Prove that $g(x,y) = \frac{x^3y}{x^2+y^2}$ with $g(0,0)=0$ is continuous
- How to prove that $\sqrt 2 + \sqrt 4$ is irrational?
- Line and plane intersection in 3D
- Prove that odd perfect square is congruent to $1$ modulo $8$
- compact Hausdorff space and continuity
- Proving pointwise convergence of series of functions
- Logic – how to write $\exists !x$ without the $\exists !$ symbol