## Investigating Some Euler Sums

So, why only odd powers? Mostly because the even powers were solved by Leonard Euler in the 18th century. Since the “mathematical toolbox” at that time did not contain the required tools, he needed 6 years to prove the validity of his deductions. Now, however, we have much more powerful tools available, as I have…

## Explore the Fascinating Sums of Odd Powers of 1/n

The goal is to get a little bit closer to the values of the zeta function (ζ(s)) and the eta function (η(s)) for some odd values of s. This insight is an expansion of two of my previous insights (Further Sums Found Through Fourier Series, Using the Fourier Series To Find Some Interesting Sums). You…

## Learn Further Sums Found Through Fourier Series

In an earlier insight, I looked at the Fourier series for some simple polynomials and what we could deduce from those series. There is a lot more to be found, however. Evaluating the Fourier series at π/2 First series In the last insight, I showed that if f(x) = x for -π<x<πand f(-π)=f(π)=0, the general …

## Using the Fourier Series To Find Some Interesting Sums

Preliminaries If f(x) is periodic with period 2p and f’(x) exists and is finite for -π<x<π, then f can be written as a Fourier series: $f(x)=\sum_{n=-\infty}^{\infty}a_{n}e^{inx}$ where $a_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-int}dt$. We shall also need Parseval’s formula. It says that for such an f we have: $\parallel f \parallel^{2}=\sum_{n=-\infty}^{\infty}\vert a_{n}\vert^{2}$ where [itex]\parallel f \parallel^{2}=\frac{1}{2\pi}\int_{-\pi}^{\pi}\vert f(t)…

## Why Road Capacity Is Almost Independent of the Speed Limit

Let us start with a familiar situation. Take a familiar part of a road, for example, the one from your home to work. How long will it take you to drive from one end of that stretch of the road to the other? At times you will have the road almost to yourself and will…

## Reflections on Technology Product Quality

On Hardware Quality It is impossible to test quality into a product. Quality must be designed into the product. The hardware designer is always responsible for the production yield. The purpose of production tests is to check production quality, not product quality. When the prototype fulfills all marketing requirements, you are only 40% done The…

## How to Manage TCP/IP In Automation or Measurement Networks

The 1990s saw the widespread adoption of network technology on two fronts: Office automation and factory automation. But while office automation soon standardized on Ethernet (for volume reasons) and TCP/IP (due to the Internet explosion), factory automation fragmented into a number of incompatible field busses (a Fieldbus is the common designation for an automation network)….

## Learn Time Synchronization Across Switched Ethernets

Now and then you come across measurement problems that are tightly associated with the notion of synchronicity, meaning that things need to happen simultaneously. The usual things that need such synchronicity are data sampling and motion control. In the case of data sampling, you need to know the value of two different quantities measured at…

## How to Write a Master Degree Thesis As a Physics Major

Your First Scientific Paper Ought to be good! Structure and Contents of a Master Degree Thesis This guide was written some years ago as a guideline for students at the Physics Master level at the University of Oslo. I had at that time been the final censor for about 30 students, and I was getting…