To solve this, we will use the Ratio Test to determine the interval of convergence for the series $S(x)$. The series is given as:

$S(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$

Step 1: General Form of the Terms

The general term $a_n$ of the series is: $a_n = (-1)^n \frac{x^{2n+1}}{2n+1}, \quad n = 0, 1, 2, \ldots$

Step 2: Apply the Ratio Test

The Ratio Test states that the series converges absolutely if: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$

Substitute $a_n$ into the ratio: $\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(-1)^{n+1} \frac{x^{2(n+1)+1}}{2(n+1)+1}}{(-1)^n \frac{x^{2n+1}}{2n+1}} \right|$

Simplify the terms: $\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{x^{2n+3}}{2n+3} \cdot \frac{2n+1}{x^{2n+1}} \right|$

This simplifies further to: $\left| \frac{a_{n+1}}{a_n} \right| = \left| x^2 \cdot \frac{2n+1}{2n+3} \right|$

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

As $n \to \infty$, $\frac{2n+1}{2n+3} \to 1$. Therefore: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = |x^2|$

For the Ratio Test, we require: $|x^2| < 1$

Step 4: Solve for $x$

$|x^2| < 1 \quad \Rightarrow \quad |x| < 1$

This implies that the series converges for $x \in (-1, 1)$.

Step 5: Check the Endpoints

We now check whether the series converges at the endpoints $x = -1$ and $x = 1$.

At $x = 1$:

Substitute $x = 1$ into the series: $S(1) = 1 - \frac{1^3}{3} + \frac{1^5}{5} - \frac{1^7}{7} + \cdots$ This is the alternating harmonic series: $\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$ The alternating harmonic series converges.

At $x = -1$:

Substitute $x = -1$ into the series: $S(-1) = -1 - \frac{(-1)^3}{3} + \frac{(-1)^5}{5} - \frac{(-1)^7}{7} + \cdots$ This simplifies to: $S(-1) = -1 + \frac{1}{3} - \frac{1}{5} + \frac{1}{7} - \cdots$ This is again the alternating harmonic series, but multiplied by $-1$. It also converges.

Step 6: Final Answer

The interval of convergence is: $[-1, 1]$

Let me know if you'd like additional details or explanations!

Related Questions:

1. How is the Ratio Test applied to series with factorial terms?
2. What happens if the Ratio Test gives a limit equal to 1?
3. Can the Alternating Series Test be used to check convergence for this series?
4. What are some other tests for determining convergence of series?
5. How does the radius of convergence relate to the interval of convergence?

Tip:

When applying the Ratio Test, always simplify the ratio $\frac{a_{n+1}}{a_n}$ as much as possible before taking the limit. This reduces the risk of algebraic errors.