This is a rough part of bigger question that I'm having troule with. Lets say I'm given some positive sequence \(\displaystyle \L t_n\), and I know that \(\displaystyle \L \sum t_n\) converges. This then imples that\(\displaystyle \L t_n \rightarrow 0\), right?
So, If I know that the above sum converges, is there a way to show that:
\(\displaystyle \L 2 \sum_{n=0}^\infty ( \sqrt{ \sum_{k=n}^{\infty} t_k }\,\, - \,\, \sqrt{\sum_{k=n+1}^{\infty} t_k})\)
converges?
So, If I know that the above sum converges, is there a way to show that:
\(\displaystyle \L 2 \sum_{n=0}^\infty ( \sqrt{ \sum_{k=n}^{\infty} t_k }\,\, - \,\, \sqrt{\sum_{k=n+1}^{\infty} t_k})\)
converges?
