Let $a _ { n } , n \geq 1$, be a sequence of positive real numbers such that $a _ { n } \longrightarrow \infty$ as $n \longrightarrow \infty$. Then which of the following are true? (A) There exists a natural number $M$ such that $$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( a _ { n } \right) ^ { M } } \in \mathbb { R }$$ (B) $$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( n ^ { 2 } a _ { n } \right) } \in \mathbb { R } .$$ (C) $$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( n a _ { n } \right) } \in \mathbb { R }$$ (D) For all positive real numbers $R$, $$\sum _ { n = 1 } ^ { \infty } \frac { R ^ { n } } { \left( a _ { n } \right) ^ { n } } \in \mathbb { R } .$$
Let $a _ { n } , n \geq 1$, be a sequence of positive real numbers such that $a _ { n } \longrightarrow \infty$ as $n \longrightarrow \infty$. Then which of the following are true?\\
(A) There exists a natural number $M$ such that
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( a _ { n } \right) ^ { M } } \in \mathbb { R }$$
(B)
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( n ^ { 2 } a _ { n } \right) } \in \mathbb { R } .$$
(C)
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( n a _ { n } \right) } \in \mathbb { R }$$
(D) For all positive real numbers $R$,
$$\sum _ { n = 1 } ^ { \infty } \frac { R ^ { n } } { \left( a _ { n } \right) ^ { n } } \in \mathbb { R } .$$