Suppose $f(1,i)>0$ is a strictly decreasing sequence of reals.
Let $f(k+1,n)=f(k,n+1)−f(k,n)$.
If $f(2m+1,n)$ is for all integers $m$, a strictly decreasing function in $n$ and $\lim_{n\rightarrow\infty}f(2m+1,n)=0$, must $\sum_{n>1}f(1,n)<\infty$?