Convergence of a series induced from given two series

Question

I could not get any counter example so I am asking this question.

Given $\frac{1}{n}\sum_{k=0}^{n}ka_k\to 0$ and $\frac{1}{n}\sum_{k=0}^{n}kb_k\to 0$, as $n\to \infty$, where $a_k,b_k \in (0,1)$.

Is it true that $\sum_{k=0}^n(a_k-b_k)\to 0$?

Thanks in advance!

Answer 1

A counterexample can be found with $a_k:=(-1)^k \frac1k$ and $b_k:=-a_k$.

Edit: This is a weak counterexmple, since only $0<\vert a_k\vert, \vert b_k\vert<1$.

Answer 2

Here is another attempt to give a counterexample.

Take $a_k:=\frac1{(k+3)\ln(k+3)}$ and $b_k:=\frac1{(k+3)^2}$. Then $a_k, b_k>0$ and $a_k\leq a_0=\frac1{3\ln{3}}<1$ and $b_k\leq b_0=\frac19<1$.

Moreover $\lim_{k\to\infty}ka_k=\lim_{k\to\infty}kb_k=0$. Thus also the arithmetic means $\frac1{n+1}\sum_{k=0}^n ka_k$ and $\frac1{n+1}\sum_{k=0}^n kb_k$ tend to $0$ for $n\to\infty$. of course this implies $\frac{1}{n}\sum_{k=0}^{n}kk_k\to 0$ and $\frac{1}{n}\sum_{k=0}^{n}kb_k\to 0$.

Finally note that $\sum_k a_k=\infty$ and that $\sum_k b_k<\infty$.