Problem

Define a sequence $\{a_n\}_{n=1}^{\infty}$ as follows. Let $a_1=1$ and $a_{n+1}=0.5a_n+\frac{1}{n}$ for $n\ge 2$. Which of the following statement is true?

(a) $\lim_{n\to\infty}a_n=1$.

(b) $\lim_{n\to\infty}a_n=\infty$.

(c) $\lim_{n\to\infty}a_n/n=0$.

(d) $\sum_{n=1}^{\infty}na_n<\infty$.

(e) None of the above statements is true.