(1)$\sum\limits_{n = 1}^\infty {\left( {\frac{1}{{\sqrt n }} - \sqrt {\ln \left( {1 + \frac{1}{n}} \right)} } \right)}$
(2)${\sum\limits_{n = 1}^\infty {\left( {\frac{1}{n} - \sin \frac{1}{n}} \right)} ^\alpha }\left( {\alpha > 0} \right)$
(3)$\sum\limits_{n = 1}^\infty {\frac{{{p^n}n!}}{{{n^n}}}} \left( {p > 0} \right)$
(4)$\sum\limits_{n = 1}^\infty {\frac{1}{{{3^{\sqrt n }}}}} $
(5)$\sum\limits_{n = 1}^\infty {\frac{1}{{\ln \left( {n + 1} \right)}}\sin \frac{1}{n}} $
(6)${\sum\limits_{n = 1}^\infty {\left( {\sqrt[n]{n} - 1} \right)} ^p}\left( {p > 0} \right)$
(7)$\sum\limits_{n = 1}^\infty {\frac{{{a^n}}}{{1 + {a^{2n}}}}} \left( {a > 0} \right)$
(8)$\sum\limits_{n = 1}^\infty {\frac{{\ln \left( {n + 2} \right)}}{{{{\left( {a + \frac{1}{n}} \right)}^n}}}} \left( {a > 0} \right)$
(9)
原文：http://www.cnblogs.com/ly758241/p/4018732.html