series

1. Prove the series are not Cauchy using the Cauchy criterion with epsilon

$b_n=\frac{x}{1}+\frac{x+1}{3}+...+\frac{x+n}{2n+1},x>1$ $c_n=\frac{1}{\sqrt1}+\frac{1}{\sqrt2}+...+\frac{1}{\sqrt n},nâ‰¥1$ I'm having troubles with the Cauchy criterion and I do not understand how to apply it when the series are not convergent. These exercises are telling me to use the Cauchy...
2. absolutely convergent series

Hello. I want to show that, for an absolutely convergent series \sum_{n=1}^{\infty}a_n, we have \left|\sum_{n=1}^{\infty}a_n\right|\leq\sum_{n=1}^{\infty}|a_n|. Let M be an positive integer. I begin with the triangle inequality \left|\sum_{n=1}^{M}a_n\right|\leq\sum_{n=1}^{M}|a_n| and taking...