The task is to prove that for sequences ${ a_n }$ and ${ A_n }$ for which the following is right: $a_n-(A_n-A_{n-1})=c_n$, where series $sum c_n $ is converging do exist such number $C$, that $a_1+a_2+a_3+…+a_n=A_n+C+O(1)$ (O(1) – O-notation here).

Came across this one during my prepartion for an exam I’m taking in a week. May you help with this one?