growth rate of sequence and series convergent

Question

Let $a_{n}$ be a real sequence, $b$ be a real number with $\lim_{n\rightarrow\infty}a_{n}=b$ but $a_{n}\neq b$ for all $n$. Let $c_{n}$ be a bounded sequence. I have shown that $n(a_{n}-b)=O(n)$ with big O notation. Consider the series $\sum_{n=1}^{\infty}\frac{(n(a_{n}-b)+c_{n})^{2}}{n(n+1)}$, is there any possibility that the series converges? If I understand correctly, $(n(a_{n}-b)+c_{n})^{2}=O(n^{2})$, so the series is eventually summing $\frac{n}{n+1}$ which diverges. But if we expand $(n(a_{n}-b)+c_{n})^{2}$ into $n^{2}(a_{n}-b)^{2}+c_{n}^{2}+2c_{n}n(a_{n}-b)$, both $n^{2}(a_{n}-b)^{2}$ and $2c_{n}n(a_{n}-b)$ diverge, may be they can cancel each other?

Answer 1

If you are asking for one example where it converges, take $a_n=\frac 1 {n^{2}}, b=0, c_n=-\frac 1 n$.