Prove that the sequence $(a_n)$ converges, where $a_n$ = $\frac{1 · 3 · 5 · · ·(2n − 1)}{2 · 4 · 6 · · ·(2n)}$. Use Monotonic Convergence Theorem.
Here is my proof:
Since $a_n = \frac{2n-1}{2n}$, then $a_{n+1} = \frac{2n-1+1}{2n+1} = \frac{2n}{2n+1}$.
Therefore, $a_n$ = $\frac{2n-1}{2n}\le\frac{2n}{2n+1}$ = $a_{n+1}$, and so the sequence, $a_n$, is increasing for all $n\in\mathbb{N}$, and is monotonic. The term $a_n$ has an upper bound (ie. 1) and a lower bound (ie. 0) for all $n$ and thus is bounded. Therefore, $a_n$ converges by the Monotonic Convergence Theorem.
Please check it over and make sure it sounds all right. If there is anything wrong, please let me know what and how I can fix it.
Best Answer
You can start by noting that the sequence is decreasing. In fact, $$ \dfrac{a_{n+1}}{a_n} = \dfrac{2n+1}{2n+2} < 1 \Rightarrow a_{n+1}< a_n. $$
Also, the sequence is clearly bounded. Since it is decreasing and positive, we have that $0 < a_n \leq a_1$.
Finally, being monotonic and bounded, the sequence is convergent.