Which of the Following Series Is Convergent, but Not Absolutely $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$

question 271

Essay

Which of the following series is convergent, but not absolutely convergent?
(a) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ (b) $\sum _ { n = 1 } ^ { \infty } \frac { \sin n } { n ^ { 2 } }$ (c) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt { n } }$ (d) $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { 2 ^ { n } + \sqrt { n } }$ (e) $\sum _ { n = 1 } ^ { \infty } \frac { 1 - 2 n } { n + 1 }$

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