关于数项级数收敛的专题讨论

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

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

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$f命题：$设$sumlimits_{n = 1}^infty  {{a_n}} $为收敛的正项级数，$left{ {n{a_n}} ight}$单调，证明：$lim limits_{n o infty } n{a_n}ln n = 0$

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$f命题：$设${a_n} > 0,sumlimits_{n = 1}^infty  {{a_n}} $收敛，证明：$sumlimits_{n = 1}^infty  {frac{{{a_n}}}{{left( {n + 1} ight){a_{n + 1}}}}} $发散

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附录1(讨论下列级数的敛散性)