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Some series of interest in probability.

n≥1 1/n   diverges:
n≥1 1/n > 1 1/x = [log x]1..∞ = ∞
So pr(n) ∝ 1/n cannot be a proper probability distribution on the positive integers, {1, 2, 3, ...}.

i≥1 1/(n(n+1))   converges to one:
1/(n(n+1)) = 1/n - 1/(n+1),
so i≥1 1/(n(n+1)) = 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ...  = 1

n≥2 1/(n loge(n))   diverges:
d/dx loge(loge(x)) = 1/(x log(x)) and,
in the integral test
2 1/(x loge(x)) = [loge(loge(x))]2..∞ = ∞

Similarly   n≥2 1/{n loge(n) loge(loge(n))}   diverges
because d/dx loge(loge(loge(x))) = 1/(x loge(x) loge(loge(x)))
etc.

n≥2 1/(n (log(n))2)   converges:
d/dx 1/loge(x) = – 1/(x (loge(x))2)
and [-1/log(x)]2..∞ = 1/log(2) is finite.

Similarly   n≥2 1/(n (log(n))p)   converges provided p>1.0,
see above.
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