Fun with inverse series

Russian: "You know what they said in the camps--'It isn't the questions that are dangerous, it's the answers'"--John Le Carre, Smiley's People

On Youtube there are channels that offer mathematical puzzles.  I saw a clip with a problem involving an inverse series.

An inverse series is something like 1+1/2+1/3+1/4+...  The interesting thing about that series is that the individual entries converge to zero, but the sum doesn't converge to a finite level, as would normally happen in such cases.  Instead, it goes on to infinity. (You can prove this through integral calculus and the natural logarithm function...)

Anyway this problem asks you to what sum the following series converges:  1+1/2+1/3 +1/4+1/5+1/6+1/8+1/9+1/10+1/12+1/15+1/16+1/18...  Notice that this isn't the sum of inverses of all natural numbers, which I just said doesn't converge; instead it's the sum of inverses of composites of powers of 2, 3 and 5.

Let's take it one step at a time.  First, the sum of inverses of powers of 2:  1+1/2+1/4+ 1/8+1/16+...  You may be able to see that this converges to 2. Then, the same for powers of 3:  1+1/3+1/9+1/27+...  That's a bit harder, but it converges to 3/2.

Now imagine the sum of inverses of composites of powers of 2 and of 3:  1+1/2+ 1/3+1/4+1/6+1/8+1/9+1/12+...When you think about it, that's actually the product of the two series I just showed you!  So it must converge to their product 2*3/2, or 3.

Now consider the series of inverse powers of 5:  1+1/5+ 1/25+1/125+...  That turns out to be 5/4.  See a pattern? A series of inverse powers of n converges to n/(n-1).  Anyhow, the solution for the convergence of the original series is similarly the product of that and the previous series:  2*3/2*5/4, in other words 15/4. (I'm so confident of the answer that I didn't watch the video to confirm it!)

Then I got thinking, what if you continued and took a series of inverses of multiples of powers of 2, powers of 3, powers of 5 and powers of 7?  That would give you 15/4*7/6, or 35/8.  Add in powers of 11, and you get 77/16.  With inverse powers of 13 too, it becomes 1001/192, et cet. ad nauseam.  As you progress through the prime numbers you'll get closer and closer to that simple inverse series of 1+1/2+1/3+1/4+... that I mentioned at the start.  Which doesn't converge, of course but goes on to infinity.

I know that a lot of people can't follow this, but it fascinates me!