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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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