关于数项级数收敛的专题讨论

$\bf命题1：$设正项级数$\sum\limits_{n = 1}^\infty {{a_n}} $发散，且${s_n} = \sum\limits_{k = 1}^n {{a_k}} $，试讨论级数$\sum\limits_{n = 1}^\infty {\frac{{{a_n}}}{{{s_n}^\alpha }}} \ $的敛散性 

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$\bf命题2：$设正项级数$\sum\limits_{n = 1}^\infty {{a_n}} $收敛，则存在发散到正无穷大的数列$\left\{ {{b_n}} \right\}$，使得级数$\sum\limits_{n = 1}^\infty {{a_n}{b_n}} $仍收敛

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$\bf命题：$设$\sum\limits_{n = 1}^\infty  {{a_n}} $为收敛的正项级数，$\left\{ {n{a_n}} \right\}$单调，证明：$\lim \limits_{n \to \infty } n{a_n}\ln n = 0$

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$\bf命题：$设${a_n} > 0,\sum\limits_{n = 1}^\infty  {{a_n}} $收敛，证明：$\sum\limits_{n = 1}^\infty  {\frac{{{a_n}}}{{\left( {n + 1} \right){a_{n + 1}}}}} $发散

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附录1(讨论下列级数的敛散性)