this post was submitted on 26 Mar 2024
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So, for starters, any exponentiation "greater than 1" is a valid candidate, in the sense that 1/(n^2), 1/(n^3), etc will all give a finite sum over infinite values of n.
From that, inverting the exponentiation "rule" gives us the "simple" examples you are looking for: 1/โn, 1/โ(โn), etc.
Knowing that
โn = n^(1/2), and so that 1/โn can be written as 1/(n^(1/2)), might help make these examples more obvious.Hang on, that's not a decreasing trend. 1/โ4 is not smaller, but larger than 1/4...?
From 1/โ3 to 1/โ4 is less of a decrease than from 1/3 to 1/4, just as from 1/3 to 1/4 is less of a decrease than from 1/(3ยฒ) to 1/(4ยฒ).
The curve here is not mapping 1/4 -> 1/โ4, but rather 4 -> 1/โ4 (and 3 -> 1/โ3, and so on).