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.