Daniel Martin

A math thing I worked out

Tue, 02 Sep 2025 06:51:45 GMT

It's somehow been in the air in a few different math places I visit online what the sum of:

1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \dots

and

\frac{1}{2} - \frac{1}{4} + \frac{1}{6} - \frac{1}{8} + \frac{1}{10} + \dots

are. Among other things, these series come up when working out the probability that with randomly chosen values for x and y in the range (0, 1) whether floor(x/y) is even or odd, or whether round(x/y) is even or odd.

Now, these are very well known series and the quite well-known answer to these are:

\begin{matrix} 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \dots & = \frac{π}{4} \\ \frac{1}{2} - \frac{1}{4} + \frac{1}{6} - \frac{1}{8} + \frac{1}{10} + \dots & = \frac{1}{2} \ln (2) \end{matrix}

The usual way that I've seen to solve the first of these ( $1 - 1 / 3 + 1 / 5 + \dots$ ) involves showing somehow that it's the Taylor series for $\arctan (x)$ evaluated at $x = 1$ ; this can be justified in detail or presented “out of thin air”. The usual way to solve the second of those involves rewriting it as:

- \frac{1}{2} \sum_{n = 1}^{\infty} \frac{(- 1)^{n}}{n}

and then working that out. For some reason these two series are approached completely separately as though the solution for one has absolutely nothing to do with the other, even though the series are obviously closely related. (I'll grant that the answers, at least for now, look completely unrelated)

Anyway, I stumbled on a way of looking at these series that makes it obvious how the two series above are related, and that as far as I can tell easily extends to the much more general case of:

\sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{h n + k}, h \in ℤ^{+}, k \in {1, 2, . . ., h}

( More math than you probably wanted to read today )

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Hey! It's an account I still have

Thu, 26 Sep 2024 19:16:58 GMT

So I guess I'm going to try things here again, now that cohost is shutting down.

I'm not sure I will use it for much, but I do think I'll probably soon be archiving and/or hiding most entries older than about 10 years, mostly because my kid doesn't need to be able to find those entries, and if he somehow wants them he can ask me about them.

Anyway though, for a first entry, here's something I'm pulling over from cohost:

A puzzle I made

So, I made a puzzle as a bit of a tribute to my dad. I have verified that solving the puzzle is possible in the sense that "if you know what went into constructing this but not the answer, it is possible to derive the answer" but of course that's no guarantee that people who are not me will naturally look at it and say "ah, these are the tools I need to wield to attack this".

Solving the puzzle requires Math, but getting to where you can figure out which Math to use first requires reading x86 assembly language. So first of, if either of those things strikes you as Intensely Not Fun, this isn't for you.

( Hard-mode puzzle, and link to slightly less-hard-mode )
( Why this puzzle )

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Deleted my livejournal!

Sat, 27 May 2017 14:21:56 GMT

Okay, I've gone and deleted my livejournal. I don't know if I'm going to resume blogging, but if I do it'll be here and not there.

In the meantime, here's a cute little result about the sum of Fibonacci numbers that I worked out while answering a question from a friend's child:

\sum_{j = 0}^{n} F_{j} = F_{n + 2}

(In case the MathML doesn't work for you, that's \sum_{j=0}^n F_j = F_{n+2})

(Using the convention that F₀=0 and F₁=F₂=1)

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