The problem requires using the ratio test to check the convergence of the series:

$\sum_{n=1}^{\infty} \frac{\sin\left(\frac{\pi}{2}n\right)}{n!}$

Solution

To apply the ratio test, we consider the terms of the series:

$a_n = \frac{\sin\left(\frac{\pi}{2}n\right)}{n!}$

The ratio test states that a series $\sum a_n$ converges absolutely if

$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$

Step 1: Compute $\frac{a_{n+1}}{a_n}$

1. For $a_{n+1}$: $a_{n+1} = \frac{\sin\left(\frac{\pi}{2}(n+1)\right)}{(n+1)!}$

2. Find $\frac{a_{n+1}}{a_n}$: $\frac{a_{n+1}}{a_n} = \frac{\sin\left(\frac{\pi}{2}(n+1)\right)}{(n+1)!} \cdot \frac{n!}{\sin\left(\frac{\pi}{2}n\right)} = \frac{\sin\left(\frac{\pi}{2}(n+1)\right)}{\sin\left(\frac{\pi}{2}n\right)} \cdot \frac{1}{n+1}$

Step 2: Analyze $\sin\left(\frac{\pi}{2}n\right)$

Notice that $\sin\left(\frac{\pi}{2}n\right)$ oscillates depending on the value of $n$:

• When $n = 1, 5, 9, \dots$, $\sin\left(\frac{\pi}{2}n\right) = 1$.
• When $n = 3, 7, 11, \dots$, $\sin\left(\frac{\pi}{2}n\right) = -1$.
• For even values of $n$, $\sin\left(\frac{\pi}{2}n\right) = 0$, which implies that some terms of the series are zero.

Step 3: Take the Limit as $n \to \infty$

Since $\sin\left(\frac{\pi}{2}(n+1)\right)/\sin\left(\frac{\pi}{2}n\right)$ oscillates and does not affect the factorial in the denominator, we focus on $\frac{1}{n+1}$.

Thus, we get: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{1}{n+1} = 0$

Conclusion

Since the limit is $0 < 1$, the series converges absolutely by the ratio test.

Would you like further details or have any questions?

Related Questions

1. How would the convergence be affected if the factorial term were removed from the denominator?
2. Can the ratio test be used for all types of series?
3. What other convergence tests could be applied to this series?
4. How does the oscillating behavior of $\sin\left(\frac{\pi}{2}n\right)$ impact the series' convergence?
5. What would happen if we replaced $\sin\left(\frac{\pi}{2}n\right)$ with another trigonometric function?

Tip: The ratio test is especially useful when terms involve factorials or exponential functions, as it simplifies the comparison of successive terms.