05/19/2010, 07:58 PM
let a_n be a strictly rising positive integer sequence.
a_n satisfies the following conditions and the question is if the conditions are reducible to a single condition.
there is also an unspecified strictly non-decreasing positive integer function f(n).
constant(k) means a constant depending on the positive integer k.
if constant(k) occurs twice it may be denoting two different functions resp constants.
sum n = 1 .. inf 1/a_n = oo
sum n = 1 .. inf 1/f(a_n) = oo
sum n = 1 .. inf f(a_n)^k /a_n - constant(k)/a_n = constant(k)
a_n > f(n).
f(n) > sum n = 1 .. n 1/a_(a_n)
... if that is even possible ...
regards
tommy1729
a_n satisfies the following conditions and the question is if the conditions are reducible to a single condition.
there is also an unspecified strictly non-decreasing positive integer function f(n).
constant(k) means a constant depending on the positive integer k.
if constant(k) occurs twice it may be denoting two different functions resp constants.
sum n = 1 .. inf 1/a_n = oo
sum n = 1 .. inf 1/f(a_n) = oo
sum n = 1 .. inf f(a_n)^k /a_n - constant(k)/a_n = constant(k)
a_n > f(n).
f(n) > sum n = 1 .. n 1/a_(a_n)
... if that is even possible ...
regards
tommy1729