In some recreational mathematics I was doing I encountered the following infinite sum: $$\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+...$$ At a glance, the terms in this sum resemble the harmonic series, $1,\frac{1}{2},\frac{1}{3},...$
My intuition told me that the above sum diverges, and a little bit of rigor shows that this is indeed the case. This led me to thinking about the idea of asymptotic density. The sequence $\frac{1}{3},\frac{1}{5},\frac{1}{7}...$ should have an asymptotic density of $1/2$ within the harmonic numbers, so it seems reasonable that its sum should diverge. So, I wanted to frame this question in a more general way. Suppose we have a real sequence $a:\mathbb{N}\rightarrow \mathbb{R} \ ; \ a:n \mapsto a(n)$ and let its sequence of partial sums be $s(n)=\sum_{k=1}^{n}a(k)$. For now, we'll assume $s(n)$ is does not converge. Now lets consider an infinite subsequence of $a$ which I'll call $a'$, which is the image of the set of natural numbers $M=\{m_1,m_2,m_3,...\}$ (where $\forall k, m_{k+1}>m_k$.) under the function $a$. I.e, $a'(k)=a(m_k).$ Is it true that if the set $M$ has a nonzero asymptotic density in $\mathbb{N}$, that the sequence of partial sums $s'(n)$ of the subsequence $a'$ will also be divergent? Or better: If $s(n)$ does converge, If the set $M$ has an asymptotic density of $0$ in $\mathbb{N}$, does $s'(n)$ necessarily converge? I've been trying to think of some sort of counterexample to either or these statemments and have not yet found one. This is an area of mathematics I do not know much about (I'm an undergraduate applied and computational mathematics student) but I am quite curious about it. Any hints that might lead me in the direction of a proof, or a neat counterexample, would be very very nice.
EDIT: To summarize, I am interested if the sum of any subsequence of density zero in a sequence with a convergent sum will also converge, or if the sum of any subsequence of nonzero density in a sequence with a divergent sum will also diverge.
EDIT 2: The user @bof produced a nice counterexample to the above. But as user @GReyes mentioned, what if we require $a(n)$ to be decreasing? Are both these ideas perhaps true in that case?