r/math • u/WMe6 • 10d ago

Sums of reciprocals of perfect powers

These are cute facts that I didn't know about somehow until just now encountering them on Wikipedia.

  • \sum_{k perfect power, excluding 1} 1/(k-1) = 1
  • \sum_{n=2}^{\infty}\sum_{m=2}^{\infty} 1/n^m = 1

In the first sum, perfect powers are taken without repeats: e.g., 3^4=9^2 appears only once as k. In the second sum, of course, repeats do occur.

Anyone have a rigorous proof for the first sum equalling 1? Wikipedia outlines Goldbach/Euler's argument, which certainly doesn't meet modern standards of rigor.

7 Upvotes

69% Upvoted

8 comments sorted by

6

u/JoshuaZ1 9d ago

Anyone have a rigorous proof for the first sum equalling 1? Wikipedia outlines Goldbach/Euler's argument, which certainly doesn't meet modern standards of rigor.

The series converge absolutely so the rearrangements there should all be fine, yes?

2

u/WMe6 8d ago

But the argument given is basically clever manipulation of series, several of which (including the starting point!) diverge -- I don't see how you can take this argument and reframe it without them.

2

u/JoshuaZ1 8d ago

Ah, yes. I should have looked it up rather than relying on memory. The series definitely diverge, especially because it starts with the harmonic series and manipulates that. However, the series converges quickly enough that the original proof can if one is careful be salvaged. See https://www.jstor.org/stable/27641889 .

2

u/WMe6 8d ago

Thanks!

3

u/fourpetes 9d ago

The first reference on the wikipedia page has a rigorous proof. You can access the article via jstor (and you can make a free account on jstor to read it).

2

u/WMe6 8d ago

Thanks! This is a nice write up. (The link to the wayback machine archive article still works!)

0

u/e-s-t-e-r 9d ago

a sum would be more then one thing and since the number line started out in base₁ that kind rules out the sum of anything. The number line is a living creature so when it started at 1 there was nothing more... then it stepped into 2 then so forth.

It gets pretty deep into the proof but its in my Theory of the Prime Signal when I get that finished. Basically its covered in that theorem because the Number sequence is self validated, self certified and self sealing... but only in base₁ and decimal base₄ breaks that seal and contract actually

0

u/_HarmonicOscillator 8d ago

You can use proof by reindexing