Let the function f(x,n,j) equal 1 when for a sequence of integers x of length n there is also a sequence of the same integer of length n+1 contained in some real number j and zero when this is not true. (For example, f(9,2,pi)=1 because there is a sequence of 2 9s at about the hundreth decimal place.)
Let the function f(x,n,j) be given in the form of a triple infinte series:
__________ n n n
f(x,n,j) = lim ∑ ∑ ∑a(x,q) + b(n,w) + c(j,e),
______ n →∞ q=1 w=1 e=1
where the varibles q, w and e are indicies of summation. Let the functions a, b and c be continuous everywhere.
(a) Is it true that
lim ∂f/∂j = k, k < ∞ ?
j→∞
(b) Is it true that for intervals (0, j) , (j+1, j+2), etc, that the following inequalities are always true?
j
∫f(x,j,n) dj <
0
2j
∫f(x,j,n) dj < ... n→∞
j
2007-02-11
20:53:04
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1 answers
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asked by
Anonymous
in
Mathematics