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To solve the problem of determining the convergence of the given series, we first need to analyze its structure.

The series is:
\[
\frac{1}{2} + \left(\frac{2}{3}\right)^2 x^2 + \left(\frac{3}{4}\right)^3 x^3 + \left(\frac{4}{5}\right)^{16} x^4 + \cdots
\]

### Step 1: Analyze the General Term
Let's denote the general term of the series as \( a_n \), where \( n \) is the index of summation. Observing the pattern, the general term appears to be:
\[
a_n = \left(\frac{n+1}{n+2}\right)^{n^2} x^n
\]

### Step 2: Test for Convergence
To test for convergence, we can apply the **Ratio Test** or the **Root Test**. The Ratio Test is usually convenient for such series.

#### Ratio Test:
The Ratio Test involves computing the limit:
\[
L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
\]
If \( L < 1 \), the series converges; if \( L > 1 \), the series diverges; if \( L = 1 \), the test is inconclusive.

Let's compute \( \frac{a_{n+1}}{a_n} \):
\[
\frac{a_{n+1}}{a_n} = \frac{\left(\frac{(n+2)}{(n+3)}\right)^{(n+1)^2} x^{n+1}}{\left(\frac{n+1}{n+2}\right)^{n^2} x^n}
\]
This simplifies to:
\[
\frac{a_{n+1}}{a_n} = \left(\frac{(n+2)^{(n+1)^2} \cdot (n+2)^{n^2}}{(n+3)^{(n+1)^2} \cdot (n+1)^{n^2}} \right) x
\]

Taking the limit as \( n \) approaches infinity:
\[
L = \lim_{n \to \infty} \left| \frac{n+2}{n+3} \right|^{2n} x = \lim_{n \to \infty} \left(1 - \frac{1}{n+3}\right)^{2n} x
\]

This limit simplifies to:
\[
L = x \cdot \lim_{n \to \infty} e^{-2} = x \cdot e^{-2}
\]

### Step 3: Conclusion
- For convergence, \( L < 1 \):
\[
x \cdot e^{-2} < 1 \implies x < e^2
\]
- For divergence, \( L > 1 \):
\[
x > e^2
\]

Thus, the series converges for \( |x| < e^2 \) and diverges for \( |x| > e^2 \).

Would you like further details on any specific part of this solution?

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Here are some related questions:

1. What is the Ratio Test and how is it applied in series convergence?
2. How do you simplify the general term in a series for convergence testing?
3. What is the Root Test, and when is it used instead of the Ratio Test?
4. How does the exponential function \( e^x \) influence series convergence?
5. What are common strategies for determining the radius of convergence?
6. How would the series behave if \( x = e^2 \)?
7. Can the alternating series test be applied to this series?
8. What are the implications if the series were an alternating series?

**Tip:** When using the Ratio Test, if the limit \( L = 1 \), consider alternative convergence tests like the Root Test or direct comparison tests.

Learn how to determine the convergence of a series using the Ratio Test and limit analysis. This detailed solution covers the general term of the series, the application of the Ratio Test, and the limit calculations involved. Suitable for advanced undergraduate students studying series convergence.

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