TIFR ECE 2017 | Question: 8

Question

Consider the two positive integer sequences, defined for a fixed positive integer $c \geq 2$

\[f(n)=\frac{1}{n}\left\lfloor\frac{n}{c}\right\rfloor, \quad g(n)=n\left\lfloor\frac{c}{n}\right\rfloor\]

where $\lfloor t\rfloor$ denotes the largest integer with value at most $t$. Which of the following statements is $\text{TRUE}$ as $n \rightarrow \infty$ ?

• A. Both sequences converge to zero
• B. The first sequence does not converge, while the second sequence converges to $0$
• C. The first sequence converges to zero, while the second sequence does not converge
• D. The first sequence converges to $1 / c$, while the second sequence converges to $0$
• E. The first sequence converges to $1 / c$, while the second sequence converges to $c$