Applications/examples of these properties?
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Here are two interesting properties on series : The first one : Let $(u_n)in(mathbb{R^+})^{mathbb{N}}$ such that $sum limits_{nge 0} u_n=+infty$ . Then there exists $(v_n)in(mathbb{R^+})^{mathbb{N}}$ such that $sum limits_{nge 0} v_n=+infty$ with $v_n underset{nto +infty}{=} o(u_n)$ . Then the second one : If $(g_k)_{kin mathbb{N}^*}$ is a strictly increasing sequence of strictly positive integers such that there exists $nu >0$ with for all $kin mathbb{N}^*$ , $(g_{k+1}-g_k)le nu (g_k-g_{k-1})$ and if $(a_n)in (mathbb{R^+})^{mathbb{N}}$ is strictly decreasing then $sum limits_{nge 1}a_n<+infty$ iff $sum limits_{kge 1}(g_{k+1}-g_k)a_{g_k}<+infty$ . Could we find applications or examples of these two properties ? Thanks in advance !
