Are you given the value of ?
The recursive relation can be written as...
(1)
The function f(x) is represented here...
There is only one 'attractive fixed point' at and because is [red line...] any possible sequence converges at . Because is [blue line...] however the convergence will be not 'monotonic' but 'oscillating'. Of course is the series diverges...
Kind regards
The general term donsn't tends to 0 if k tends to infinity, so that the series diverges... it is interesting to valuate the series where the satisfy the recursive relation...
(1)
The (1) can be written as...
(2)
The function f(x) is represented here...
There is only one 'attractive fixed point' at and it is clear that is...
(3)
... no matter which is and the convergence is 'monotonic', so that the series has only positive or negative terms. The (3) tells us that the series may converge, but the question is: does it converge or not?... the question isn't trivial... for example the ratio test fails because is...
(4)
Does someone have some 'good idea'?...
Kind regards
Perhaps I am misunderstanding what you are trying to prove. I thought you were trying to determine whether or not the series converged and prove it.
One of the very first theorems about infinite series is that if the series converges, then the sequence must go to 0. The contrapositive of that is that if the sequence does not go to 0, then the series does not converge.
Now, what exactly are you trying to prove?