Rational Number Expressible as Sum of Reciprocals of Distinct Squares/Mistake

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Source Work

1992: David Wells: Curious and Interesting Puzzles:

The Puzzles:
Egyptian Fractions


Mistake

The sum of the series $1 + 1 / 2^2 + 1 / 3^2 + 1 / 4^2 \ldots = \pi^2 / 6$, so the sum of different Egyptian fractions whose denominators are squares cannot exceed $\pi^2 / 6$, but might equal, for example, $\frac 1 2$.


Correction

It is implicit that $1$ is not included in the set of Egyptian fractions.


We have that:

\(\ds \dfrac {\pi^2} 6\) \(\approx\) \(\ds 1.6449 \ldots\)
\(\ds \leadsto \ \ \) \(\ds \dfrac {\pi^2} 6 - 1\) \(\approx\) \(\ds 0.6449 \ldots\)


Hence the sentence should end:

... cannot exceed $\pi^2 / 6 - 1$, but might equal, for example, $\frac 1 2$.


Sources