Let $(a_k)_{k \in \mathbb{N}} \subset ]0,+\infty[$, and assume that $a_k \to 0$ as $k$ goes to $+\infty$, but $a_k \notin \ell^1(\mathbb{N})$.

It is easy to prove that we can extract a subsequence $(a_{k_n})_{n \in \mathbb{N}} \in \ell^1(\mathbb{N})$; for this it is enough to choose $k_n$ going to $+\infty$ "fast enough".

I wonder if the following implication is true: $(a_{k_n})_{n \in \mathbb{N}} \in \ell^1(\mathbb{N})$ $\Rightarrow$ $(\frac{1}{k_n})_{n \in \mathbb{N}} \in \ell^1(\mathbb{N})$ ?

I also wonder if there is a way to reverse this implication, by doing an additional summability assumption like $(a_k)_{k \in \mathbb{N}} \in \ell^p(\mathbb{N})$?

Your Answer

 

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.