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We are asked to find the radius and interval of convergence for the given series:

\[
\sum_{k=1}^{\infty} \frac{x^k}{k(k+2)}
\]

To find the radius and interval of convergence, we will use the **Ratio Test** to analyze the series.

### Step 1: Ratio Test Setup
For a series of the form:

\[
\sum_{k=1}^{\infty} a_k x^k
\]

The Ratio Test gives the radius of convergence \(R\) by examining the limit:

\[
L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|
\]

If \(L = \frac{1}{R}\), then:
- The series converges when \(|x| < R\).
- The series diverges when \(|x| > R\).

### Step 2: Find the Ratio

In this case, the general term of the series is:

\[
a_k = \frac{1}{k(k+2)}
\]

Thus, we will calculate the ratio:

\[
\frac{a_{k+1}}{a_k} = \frac{\frac{1}{(k+1)((k+1)+2)}}{\frac{1}{k(k+2)}}
= \frac{k(k+2)}{(k+1)(k+3)}
\]

Now, let's simplify this expression and find the limit as \(k \to \infty\):

\[
\lim_{k \to \infty} \frac{k(k+2)}{(k+1)(k+3)} = \lim_{k \to \infty} \frac{k^2 + 2k}{k^2 + 4k + 3} = \lim_{k \to \infty} \frac{1 + \frac{2}{k}}{1 + \frac{4}{k} + \frac{3}{k^2}} = 1
\]

The ratio approaches 1 as \(k\) becomes large.

### Step 3: Conclusion
Since the limit approaches 1, the Ratio Test is inconclusive. To determine the exact radius of convergence, we need to check for absolute and conditional convergence using other convergence tests, but from the structure of the series, we can predict that **the radius of convergence is infinite** because the ratio of successive terms does not depend on \(x\).

Thus:
- **Radius of Convergence:** \(R = \infty\).
- **Interval of Convergence:** \((-\infty, \infty)\).

Would you like further details, or do you have any questions?

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### 5 Related Questions:
1. How does the Ratio Test apply to different types of power series?
2. What happens if the Ratio Test gives a limit of less than 1?
3. Can the root test also be used to find the radius of convergence?
4. How do we determine convergence at the endpoints of an interval of convergence?
5. What is the difference between absolute and conditional convergence in series?

### Tip:
For more complex series, always check if the Ratio Test leads to a clear conclusion. If not, alternative tests like the Root Test or specific convergence tests may be more effective.

Find the radius of convergence and the interval of convergence for the series sum from k=1 to infinity of (x^k) / (k(k+2)).

Learn how to find the radius and interval of convergence for the series sum from k=1 to infinity of (x^k) / (k(k+2)). This detailed solution uses the Ratio Test to determine the convergence properties of the series. Suitable for undergraduate students.

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