Using D'Alembert's ratio test, determine the convergence
Using D'Alembert's ratio test, determine the convergence of: $$ \sum_{n=1}^{\infty} \frac{(2n-1)!!}{(2n)!!} \cdot \frac{1}{2^{n+1}}. $$
Explanation of Method
The double factorials $(2n-1)!!$ and $(2n)!!$ each shift by a single new factor when $n\to n+1$ — namely $(2n+1)$ and $\frac1{2n+2}$ respectively — so their ratio simplifies cleanly. Combined with the extra factor of $\frac12$ from the $2^{n+1}$ denominator, the whole ratio reduces to a simple rational function of $n$ whose limit is $\frac12$, safely inside the convergent range.
- Simplify the ratios of consecutive double factorials.
- Combine with the exponential factor and take the limit.
- Apply the ratio test to conclude convergence.
Step 1 · Simplify the double-factorial ratios
With $a_n=\dfrac{(2n-1)!!}{(2n)!!}\cdot\dfrac1{2^{n+1}}$, using $(2n+1)!!=(2n+1)(2n-1)!!$ and $(2n+2)!!=(2n+2)(2n)!!$:
$$ \frac{a_{n+1}}{a_n} = \frac{(2n+1)!!}{(2n-1)!!}\cdot\frac{(2n)!!}{(2n+2)!!}\cdot\frac12 = (2n+1)\cdot\frac1{2n+2}\cdot\frac12. $$
Step 2 · Combine and take the limit
$$ \frac{a_{n+1}}{a_n} = \frac{2n+1}{2(2n+2)} = \frac{2n+1}{4n+4} \longrightarrow \frac{2}{4} = \frac12. $$
Step 3 · Apply the ratio test
Since $L=\dfrac12<1$, by D'Alembert's ratio test $\sum_{n=1}^{\infty}\dfrac{(2n-1)!!}{(2n)!!}\cdot\dfrac1{2^{n+1}}$ converges.
Summary Table
| Quantity | Value |
|---|---|
| Ratio $a_{n+1}/a_n$ | $\dfrac{2n+1}{4n+4}$ |
| $L=\lim a_{n+1}/a_n$ | $\dfrac12$ |
| Conclusion | Converges |
Additional Notes
- The double-factorial ratio $\dfrac{(2n-1)!!}{(2n)!!}$ is closely related to the central binomial coefficient: it equals $\dfrac{(2n)!}{2^{2n}(n!)^2}=4^{-n}\binom{2n}n$, connecting this series to combinatorial identities and to the asymptotic $\binom{2n}n\sim\frac{4^n}{\sqrt{\pi n}}$.
- Numerically the ratios $\frac{a_2}{a_1}=0.375$, $\frac{a_3}{a_2}\approx0.417$ approach $\frac12$ from below, consistent with the exact formula $\frac{2n+1}{4n+4}$ increasing toward its limit as $n$ grows.
- This series demonstrates that double factorials, despite their unusual definition, obey the same ratio-test machinery as ordinary factorials once their recursive structure ($( 2n+1)!!=(2n+1)(2n-1)!!$, etc.) is exploited.
Result
The series converges: