TIFR-2015-Maths-A-9

Question

Let $\left\{a_{n}\right\}$ be a sequence of real numbers such that $|a_{n+1}-a_{n}|\leq \frac{n^{2}}{2^{n}}$ for all $n \in \mathbb{N}$. Then

• A. The sequence $\left\{a_{n}\right\}$ may be unbounded.
• B. The sequence $\left\{a_{n}\right\}$ is bounded but may not converge.
• C. The sequence $\left\{a_{n}\right\}$ has exactly two limit points.
• D. The sequence $\left\{a_{n}\right\}$ is convergent.