why Beta(1,1) is an improper prior While I am looking for the sun-rise problem, in part of prior selection they said that ...
in a section of Beta distribution https://en.wikipedia.org/wiki/Beta_distribution#Rule_of_succession

Jaynes[53] questions (for the uniform prior Beta(1,1)) the use of these formulas for the cases s = 0 or s = n because the integrals do not converge (Beta(1,1) is an improper prior for s = 0 or s = n). 

in proof of rule of succession which is using Beta(1,1) as its prior https://en.wikipedia.org/wiki/Rule_of_succession

This improper prior is 1/(p(1 − p)) for 0 ≤ p ≤ 1 and 0 otherwise.[5] 

Where the p is an unknown parameter for the probability of success.
Though I don't know where 1/(p(1 − p)) comes from.

Beta(1,1) is  Uniform(0,1) which is proper prior.
Apparently, it is integrated into one, thus a proper prior.
Question: why prior Beta(1,1) in proof for the rule of succession is improper?
 A: The Wikipedia passage

Jaynes [53] questions (for the uniform prior Beta(1,1)) the use of
  these formulas for the cases $s = 0$ or $s = n$ because the integrals do
  not converge (Beta(1,1) is an improper prior for $s = 0$ or $s = n$). In
  practice, the conditions $0<s<n$ necessary for a mode to exist between
  both ends for the Bayes prior are usually met, and therefore the Bayes
  prior (as long as $0 < s < n$) results in a posterior mode located
  between both ends of the domain.

is clearly a typo as the rest of the section

Haldane's prior probability (Beta(0,0))
The Beta(0,0) distribution was proposed by J.B.S. Haldane,[64] who
  suggested that the prior probability representing complete uncertainty
  should be proportional to $p^{−1}(1−p)^{−1}$ (...)

calls the Bayes-Laplace prior a Beta(1,1) distribution, Haldane's prior a Beta(0,0) [improper] distribution, and Jeffrey's middle ground a Beta(1/2,1/2) distribution.
Ànother sentence that seems to be wrong is

According to Jaynes [53] the main problem with the rule of succession
  is that it is not valid when $s=0$ or $s=n$ (see rule of succession,
  for an analysis of its validity).

since Jeffreys' prior, which is supported by Jaynes, has no issue when $s=0$ or $s=n$.
