$$b_n=\frac{x}{1}+\frac{x+1}{3}+\frac{x+2}{5}+...+\frac{x+n}{2n+1},x\gt1$$ $$c_n=\frac{1}{\sqrt1}+\frac{1}{\sqrt2}+...+\frac{1}{\sqrt n},n\geq1$$ 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 criterion $$\lvert b_{n+p}-b_{n}\rvert<ε$$.
Is the absolute value inside always positive? I've learned in school that for every negative value inside the absolute value symbol it changes to a positive value. Ex: $\lvert-9\rvert$=$9$ or $\lvert -2x-5 \rvert$=$2x+5$.
However since these series are not convergent I need to prove that by using the criterion with epsilon.How do you apply that? If you apply the criterion can you reduce terms if they are inside the absolute value symbol?