If I remember my series analysis math classes correctly: technically, summing a decreasing trend up to infinity will give you a finite value if and only if the trend decreases faster than the function/curve x -> 1/x.
Great. Can you give me example of decreasing trend slower than that function curve?, where summation doesn't give finite value? A simple example please, I am not math scholar.
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/√n, 1/√(√n), etc.
Knowing that √n = n^(1/2), and so that 1/√n can be written as 1/(n^(1/2)), might help make these examples more obvious.
Hang on, that's not a decreasing trend. 1/√4 is not smaller, but larger than 1/4...?
From 1/√3 to 1/√4 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/(3²) to 1/(4²).
The curve here is not mapping 1/4 -> 1/√4, but rather 4 -> 1/√4 (and 3 -> 1/√3, and so on).
If I remember my series analysis math classes correctly: technically, summing a decreasing trend up to infinity will give you a finite value if and only if the trend decreases faster than the function/curve
x -> 1/x.Great. Can you give me example of decreasing trend slower than that function curve?, where summation doesn't give finite value? A simple example please, I am not math scholar.
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/√n, 1/√(√n), etc.
Knowing that
√n = n^(1/2), and so that 1/√n can be written as 1/(n^(1/2)), might help make these examples more obvious.Hang on, that's not a decreasing trend. 1/√4 is not smaller, but larger than 1/4...?
From 1/√3 to 1/√4 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/(3²) to 1/(4²).
The curve here is not mapping 1/4 -> 1/√4, but rather 4 -> 1/√4 (and 3 -> 1/√3, and so on).