Given x0>0 and a>0, show that there exists one and only one sequence {x0, x1, x2,...) such that x_n = \sum_{j=n+1}^{\infty} x_j^a
1well i think...............take two sequences..........
Xn=A...........and also Xn=B........
after doing sum mathematical stuff............
try to get A=B...........
thts all
by dis we can show tht it is unique.............
21yes sir k is a function of m and n but it is always greater than zero for all m,n>0 this was true only if a is a positive integer..
by the way hope this will suffice..
if we prove f(x)= x+ xa is monotonic for all x>0 (where a is a positive real number) then also we prove that x(n+1) is unique...f"(x)>0 for all x>0
62there is one small problem shubhodeep..
it is not very clear why x-y is a factor of x^a-y^a..
since a is any real number not an integer... and what is the guarantee that the other factor k whihc you have taken is not zero for any m and n which are not equal
(since k is not a constant but a function of m and n!)
21Giving a try to this one...
x(n)=Xa(n+1) + xa(n+2) ......
x(n+1) = xa(n+2)....
so x(n) = x(n+1) + xa(n+1)
as prophet sir said, if we prove x(n) obtained from this relation is unique we r done..
so we have to prove that the function f(x)= x + xa does not take same value at two distinct points in its domain...at least for x>0 because we have x(j)>0
to prove that by contradiction method lets take two distinct point m and n in its domain (m,n>0) for which we hace f(m)= f(n)
so we have m-n + ma - na=o
or (m-n)(k)=0 (where k>0 as m,n>0)
only possibility m=n
so this is one-to-one function...so proved...
341Well priyam has written x_n = x_{n+1} + x_{n+1}^a
Now, if we prove that xn+1 obtained from this relation is unique, we will have established that the sequence is also unique.
So, now consider the function f(x) = x + x^a
Is there a way to prove that this function is one-one?
62Akand you have given a very good reasoning..
try to go ahead and prove what you said.. you are quite close :)
33If Xn+1 is unique then Xn is unique.. so have to show x∞ unique.... (lekin x∞ ... :P)
341wow! absolutely correct sir. (I think I am now justfied in calling you sir)
39how to prove that a sequence is unique? gimme a hint and i think i can carry off
