Exercise 3, § 7

Let $(x_n)_{n \geq 0}$ be a decreasing sequence of finite real numbers $\geq 0$. This sequence is summable in $\mathbf{R}$ if and only if the sequence $(2^n x_{2^n})_{n \geq 0}$ is summable in $\mathbf{R}$ ("Cauchy's condensation test"). \* Deduce that the series whose general term is $1/n^\alpha$ is convergent if $\alpha > 1$ and not convergent if $0 < \alpha \leq 1$. \*