The series given in the image is:

$\sum_{n=1}^{\infty} \frac{a^n}{(n!)^{1/n}}, \quad \text{with} \ a > 0.$

To study the convergence of this series, we can apply several tests for series convergence, such as the Ratio Test or Root Test, as these are useful for series with factorials and powers.

Root Test (Cauchy’s Root Test)

The Root Test is particularly useful in this case because of the $(n!)^{1/n}$ term. According to the Root Test, we compute:

$L = \lim_{n \to \infty} \sqrt[n]{\left| \frac{a^n}{(n!)^{1/n}} \right|}.$

Let's simplify this expression step by step:

$L = \lim_{n \to \infty} \left( \frac{a^n}{(n!)^{1/n}} \right)^{1/n} = \lim_{n \to \infty} \frac{a}{(n!)^{1/n^2}}.$

As $n \to \infty$, Stirling’s approximation for factorials ($n! \sim \sqrt{2\pi n} \left( \frac{n}{e} \right)^n$) implies that:

$(n!)^{1/n^2} \sim \left( \frac{n}{e} \right)^{1/n} = 1.$

Thus, we have:

$L = \lim_{n \to \infty} a \cdot 1 = a.$

Conclusion of the Root Test:

• If $a < 1$, then $L < 1$, and the series converges.
• If $a = 1$, then $L = 1$, and the test is inconclusive.
• If $a > 1$, then $L > 1$, and the series diverges.

Thus, the series converges for $a < 1$ and diverges for $a > 1$.

Would you like further details or have any specific questions?

Here are 5 related questions to explore:

1. How would the series behave if $a = 1$?
2. Can the Ratio Test be applied for similar types of series?
3. How does Stirling's approximation help in understanding factorials in series?
4. What other series convergence tests can be applied to this series?
5. How does the Root Test compare to the Ratio Test in terms of applicability?

Tip: When factorials are involved in a series, Stirling’s approximation often simplifies the analysis for large $n$.