= Solution
Absorb the finitely many exceptional initial terms in the <Big O notation> assumption into a constant $C>0$, so that $|a_n|\leq Cn$ for every $n\geq1$. This is possible because each initial term is finite and each index is positive. We prove the stronger bound $|{}^kb_n|\leq Cn$ for every $k,n\geq1$.
The case $k=1$ is the chosen bound. If it holds at $k-1$, the <triangle inequality> gives
$$
|{}^kb_n|\leq\frac1n\sum_{j=1}^n|{}^{k-1}b_j|\leq\frac Cn\sum_{j=1}^n j=\frac{C(n+1)}2\leq Cn.
$$
<Mathematical induction> completes the proof. Thus \b[${}^kb_n=O(n)$ for every fixed $k$]; in fact the same constant works for all the iterated <Cesàro means>. This is the <linear-growth bound for iterated Cesàro means>.
Back to article page
