find the radius of convergence of ((-1)^(n-1)*(x^n))/(n^3)

To find the radius of convergence \( R \) of the power series

\[
\sum_{n=1}^{\infty} \frac{(-1)^{n-1}x^n}{n^3}
\]

we can apply the **Root Test** or **Ratio Test**. Let's use the **Root Test** for this case, which is given by:

\[
\frac{1}{R} = \lim_{n \to \infty} \sqrt[n]{|a_n|}
\]

where \( a_n = \frac{(-1)^{n-1}}{n^3} \). Since \( (-1)^{n-1} \) only affects the sign and not the magnitude, we have:

\[
|a_n| = \frac{1}{n^3}
\]

Now, applying the root test:

\[
\frac{1}{R} = \lim_{n \to \infty} \sqrt[n]{\left|\frac{x^n}{n^3}\right|}
= \lim_{n \to \infty} \sqrt[n]{\frac{|x|^n}{n^3}}
\]

This simplifies to:

\[
\frac{1}{R} = \lim_{n \to \infty} \frac{|x|}{n^{3/n}}
\]

As \( n \to \infty \), \( n^{3/n} \to 1 \), so:

\[
\frac{1}{R} = |x|
\]

Thus, the radius of convergence \( R \) is:

\[
R = 1
\]

### Conclusion:
The radius of convergence of the series is \( R = 1 \).

Would you like more details on the convergence tests or behavior at the endpoints? 

Here are some related questions for deeper understanding:
1. What is the behavior of the series at \( x = 1 \) and \( x = -1 \)?
2. How does the Ratio Test compare to the Root Test for this problem?
3. Can alternating series affect convergence at the endpoints?
4. How does the general term \( n^3 \) in the denominator influence the convergence rate?
5. What would happen if the exponent on \( n \) were different, say \( n^2 \) or \( n^4 \)?

**Tip:** The Root Test is often easier for series with powers like \( n^k \), as it simplifies faster than the Ratio Test.

Find the radius of convergence of ((-1)^(n-1)*(x^n))/(n^3).

Learn how to find the radius of convergence for the power series (-1)^(n-1) * (x^n) / (n^3) using the Root Test. This solution covers key concepts like convergence tests and radius of convergence, suitable for first-year calculus students.

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