Let $\mathbb { R } ^ { + } = \{ x \in \mathbb { R } : x \geq 0 \}$. For $x \in \mathbb { R } ^ { + }$, denote by $\operatorname{FRAC}( x )$ the fractional part of $x$, i.e., $x - n$ where $n$ is the largest integer that is less than or equal to $x$. Consider the series $\sum _ { n = 1 } ^ { \infty } \frac { \operatorname { FRAC } ( x / n ) } { n }$. Pick the correct statement(s) from below. (A) The above series converges for all $x \in \mathbb { R } ^ { + } - \mathbb { Z }$. (B) The above series diverges for some non-negative integer $x$. (C) The above series defines a continuous function in a neighbourhood of $\frac { 1 } { 2 }$. (D) The above series defines a continuous function in a neighbourhood of 1.
Let $\mathbb { R } ^ { + } = \{ x \in \mathbb { R } : x \geq 0 \}$. For $x \in \mathbb { R } ^ { + }$, denote by $\operatorname{FRAC}( x )$ the fractional part of $x$, i.e., $x - n$ where $n$ is the largest integer that is less than or equal to $x$. Consider the series $\sum _ { n = 1 } ^ { \infty } \frac { \operatorname { FRAC } ( x / n ) } { n }$. Pick the correct statement(s) from below.\\
(A) The above series converges for all $x \in \mathbb { R } ^ { + } - \mathbb { Z }$.\\
(B) The above series diverges for some non-negative integer $x$.\\
(C) The above series defines a continuous function in a neighbourhood of $\frac { 1 } { 2 }$.\\
(D) The above series defines a continuous function in a neighbourhood of 1.