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
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Egyptian Fractions