I am thinking again about the real sequence $\{a_n\}_{n\ge1}$ which decays faster than any algebraic speed, that is, $\lim_{n\to \infty}n^pa_n = 0$ for every integer $p$. Actually, $a_n$ can be treated as the Fourier sequence of a smooth periodic function. Assume now the following two equations hold for every integer $k\ge 0$:

  1. $\sum_{n\ge 1}n^k(n+1)^k(\frac{n+1}{n}-\frac{n}{n+1})a_na_{n+1} = 0$;
  2. $\sum_{n\ge 1}n^k(n+2)^k(\frac{n+2}{n}-\frac{n}{n+2})a_na_{n+2} = 0$.

In this case, can we conclude that $a_na_{n+1} = 0$ and $a_na_{n+2} = 0$? I already know from the discussion in A question on a real sequence that only one of the above assumption does not lead to the conclusion. I am not sure what will happen if we combine these two conditions together.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.