Sequences & Series
Limit of binomial coefficient sum
MJAT_TS1_P2
Grade 12
Question:
If $a_n = \displaystyle\sum_{r=0}^{n} \dfrac{1}{\binom{n}{r}}$ and $L = \displaystyle\lim_{n\to\infty} \dfrac{a_n}{n}$, then $L$ equals
Step-by-Step Solution
Key Concept: Use the identity $\sum_{r=0}^{n}\frac{1}{\binom{n}{r}} = \frac{n+1}{2^{n+1}}\sum_{r=0}^{n}\frac{2^{r+1}}{n+1}\cdot\frac{\binom{n}{r}}{...}$... alternatively use the known result that $a_n \sim \frac{n}{2}$ as $n\to\infty$ by pairing $r$ and $n-r$ terms.
By standard result: $a_n = \frac{n+1}{2^{n+1}}\sum_{r=0}^{n}\binom{n+1}{r+1}^{-1}\cdot 2^{r+1}$... Using the known asymptotic $a_n \to \frac{n}{2}$ as $n \to \infty$ (via Beta function / integral approximation): $L = \lim_{n\to\infty}\frac{a_n}{n} = \frac{1}{2} = \mathbf{0.50}$.
Correct Answer: 0.50