Determine Whether Each of the Following Series Converges $\sum _ { n = 1 } ^ { \infty } \frac { 2 + ( - 1 ) ^ { n } } { 2 ^ { n } }$

Short Answer

Determine whether each of the following series converges. Justify your answer by specifying which test you are using and showing any necessary computation.
(a) $\sum _ { n = 1 } ^ { \infty } \frac { 2 + ( - 1 ) ^ { n } } { 2 ^ { n } }$
(b) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } + 1 } { 3 n ^ { 3 } - n + 2 }$
(c) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 + \sin n } { n } \right) ^ { 2 }$
(d) $\sum _ { n = 1 } ^ { \infty } \frac { n ! } { 1 \cdot 3 \cdot 5 \cdots \cdots ( 2 n - 1 ) }$
(e) $\sum _ { n = 1 } ^ { \infty } \frac { n ! } { n ^ { n } }$
(f) $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln n } { n } \right) ^ { n }$
(g) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n \sqrt { 1 + \ln n } }$
(h) $\sum _ { n = 1 } ^ { \infty } \frac { \ln n } { ( n + 1 ) ^ { 3 } }$
(i) $\sum _ { n = 1 } ^ { \infty } 3 ^ { n } \sin \left( \frac { \pi } { 4 ^ { n } } \right)$
(j) $\sum _ { n = 1 } ^ { \infty } \tan ^ { - 1 } n$
(k) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { \ln n } { n }$
(l) $\sum _ { n = 1 } ^ { \infty } \ln \left( 1 + \frac { 1 } { n } \right)$
(m) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } n ! } { 2 \cdot 5 \cdot 8 \cdot \ldots ( 3 n - 1 ) }$
(n) $\sum _ { n = 1 } ^ { \infty } \frac { \sqrt { n + 1 } - \sqrt { n } } { n }$