The given problem asks to find the radius of convergence and the interval of convergence for the series:

$\sum_{k=0}^{\infty} \frac{x^k}{k+1}$

Step 1: Analyze the Series

We need to use the Ratio Test to determine the radius of convergence. The ratio test examines the limit:

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

For the given series, the general term $a_k$ is:

$a_k = \frac{x^k}{k+1}$

Now, applying the Ratio Test:

$\frac{a_{k+1}}{a_k} = \frac{\frac{x^{k+1}}{k+2}}{\frac{x^k}{k+1}} = \frac{x \cdot (k+1)}{k+2}$

Simplifying:

$\frac{a_{k+1}}{a_k} = x \cdot \frac{k+1}{k+2}$

As $k \to \infty$:

$\lim_{k \to \infty} \frac{k+1}{k+2} = 1$

Thus, the limit of the ratio becomes:

$L = |x| \cdot 1 = |x|$

Step 2: Apply the Ratio Test

For convergence, the Ratio Test requires:

$L < 1 \quad \Rightarrow \quad |x| < 1$

Thus, the radius of convergence is:

$R = 1$

Step 3: Determine the Interval of Convergence

We know the series converges for $|x| < 1$. Now, we check the endpoints $x = -1$ and $x = 1$.

1. At $x = 1$:

   The series becomes:

   $\sum_{k=0}^{\infty} \frac{1^k}{k+1} = \sum_{k=0}^{\infty} \frac{1}{k+1}$

   This is the harmonic series, which diverges.

2. At $x = -1$:

   The series becomes:

   $\sum_{k=0}^{\infty} \frac{(-1)^k}{k+1}$

   This is the alternating harmonic series, which converges by the alternating series test.

Final Answer:

• The radius of convergence is $R = 1$.
• The interval of convergence is $[-1, 1)$, where the series converges at $x = -1$ but not at $x = 1$.

Would you like more details or explanations on any specific part?

Here are 5 related questions to deepen your understanding:

1. How does the Ratio Test help in determining the radius of convergence?
2. What is the significance of testing endpoints when finding the interval of convergence?
3. Can alternating series always be expected to converge at endpoints?
4. How does the convergence behavior of the harmonic series differ from that of the alternating harmonic series?
5. How can you apply the Ratio Test to more complex series involving factorials or powers?

Tip: Always apply the Ratio Test carefully and remember to check the endpoints separately for the interval of convergence.