If , then the series converges absolutely and that means that and for k 'large enough' is where . Now if we suppose that and set it will be and for k 'large enough' where so that ...
Kind regards
Hi all,
I'm trying to solve the following 2 questions , can anyone help?
1. Let for k . Show that there exists a sequence 0 such that
2.Let .Show that there exists a sequence such that .
For the first part , I think for is suitable. Can anyone give a hint for the second part?
Unfortunately that is not true. There exist convergent series of positive numbers, such that the condition does not hold for all 'large enough' n, for any . For example, if , then converges (by the integral test), but for all .
There is a standard trick to construct this example. Let denote the tail of the series , in other words . Then as . Let . Clearly as .
For the clever proof that converges, see here.