Thread: A guess of series

Whether the series $\displaystyle \sum_{n=1}^{\infty}a_{n}$ convergence or not ,
can we get the conclusion $\displaystyle \sum_{n=1}^{\infty}o(a_{n}) = o(\sum_{n=1}^{\infty}a_{n})$?
can someone give a prove of it or a counter-example?

Originally Posted by Xingyuan

Whether the series $\displaystyle \sum_{n=1}^{\infty}a_{n}$ convergence or not ,
can we get the conclusion $\displaystyle \sum_{n=1}^{\infty}o(a_{n}) = o(\sum_{n=1}^{\infty}a_{n})$?
can someone give a prove of it or a counter-example?

What is $\displaystyle o(\sum_{n=1}^{\infty}a_{n})$ supposed to mean? (the little o notation is indexed by a variable, like $\displaystyle o_n(a_n)$ in the left hand side, so what should the variable be here??)