A strange take on the harmonic series

It is well known that the harmonic series

1 + ½ + ⅓ + ¼ + …

diverges. But if you take the denominators as numbers in base 11 or higher, the series converges [1].

I wonder what inspired this observation. Maybe Brewster was bored, teaching yet another cohort of students that the harmonic series diverges, and let his mind wander.

Proof

Let f(n) be the function that takes a positive integer n, writes it in base 10, then reinterprets the result as a number in base b where b > 10. Brewster is saying that the sum of the series 1/f(n) converges.

To see this, note that the first 10 terms are less than or equal to 1. The next 100 terms are less than 1/b. The next 1000 terms are less than 1/b², and so on. This means the series is bounded by the geometric series 10 (10/b)^m.

Python

Incidentally, despite being an unusual function, f is very easy to implement in Python:

   def f(n, b): return int(str(n), b)

Citation

Brewster’s note was so brief that I will quote it here in full.

The [harmonic series] is divergent. But if the denominators of the terms are read as numbers in scale 11 or any higher scale, the series is convergent, and the sum is greater than 2.828 and less than 26.29. The convergence is rather slow. I estimate that, to find the last number by direct addition, one would have to work out 10⁹⁰ terms, to about 93 places of decimals.

[1] G. W. Brewster. An Old Result in a New Dress. The Mathematical Gazette, Vol. 37, No. 322 (Dec., 1953), pp. 269–270.

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