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(1)$\sum\limits_{n = 1}^\infty  {\left( {\frac{1}{{\sqrt n }} - \sqrt {\ln \left( {1 + \frac{1}{n}} \right)} } \right)}$

1

(2)${\sum\limits_{n = 1}^\infty  {\left( {\frac{1}{n} - \sin \frac{1}{n}} \right)} ^\alpha }\left( {\alpha  > 0} \right)$

1

(3)$\sum\limits_{n = 1}^\infty  {\frac{{{p^n}n!}}{{{n^n}}}} \left( {p > 0} \right)$

1

(4)$\sum\limits_{n = 1}^\infty  {\frac{1}{{{3^{\sqrt n }}}}} $

1

(5)$\sum\limits_{n = 1}^\infty  {\frac{1}{{\ln \left( {n + 1} \right)}}\sin \frac{1}{n}} $

1

(6)${\sum\limits_{n = 1}^\infty  {\left( {\sqrt[n]{n} - 1} \right)} ^p}\left( {p > 0} \right)$

1

(7)$\sum\limits_{n = 1}^\infty  {\frac{{{a^n}}}{{1 + {a^{2n}}}}} \left( {a > 0} \right)$

1

(8)$\sum\limits_{n = 1}^\infty  {\frac{{\ln \left( {n + 2} \right)}}{{{{\left( {a + \frac{1}{n}} \right)}^n}}}} \left( {a > 0} \right)$

1

(9)