The problem is to evaluate the indefinite integral

$\int \ln(1+x) \, dx$

as a power series, and then determine the radius and interval of convergence.

Step 1: Represent $\ln(1+x)$ as a power series

The Taylor series expansion for $\ln(1+x)$ around $x = 0$ is:

$\ln(1+x) = \sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n}, \quad |x| < 1.$

So, the logarithm function expands to:

$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots$

Step 2: Integrate term by term

Now, integrate the series term by term:

$\int \ln(1+x) \, dx = \int \left( x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots \right) dx.$

Integrating each term:

$\int x \, dx = \frac{x^2}{2}, \quad \int \frac{x^2}{2} \, dx = \frac{x^3}{6}, \quad \int \frac{x^3}{3} \, dx = \frac{x^4}{12}, \dots$

Thus, the power series for the integral is:

$\int \ln(1+x) \, dx = \frac{x^2}{2} - \frac{x^3}{6} + \frac{x^4}{12} - \frac{x^5}{20} + \dots + C,$

or equivalently:

$\int \ln(1+x) \, dx = \sum_{n=1}^\infty (-1)^{n+1} \frac{x^{n+1}}{n(n+1)} + C,$

where $C$ is the constant of integration.

Step 3: Radius and Interval of Convergence

The series representation of $\ln(1+x)$ converges for $|x| < 1$, so the power series for the integral also converges within this interval. At the endpoints ($x = \pm 1$):

• For $x = 1$, the series diverges (harmonic series test).
• For $x = -1$, the series alternates but does not converge absolutely.

Thus, the radius of convergence is $R = 1$, and the interval of convergence is:

$(-1, 1).$

Final Answer:

1. Power series for the integral:

$\int \ln(1+x) \, dx = \sum_{n=1}^\infty (-1)^{n+1} \frac{x^{n+1}}{n(n+1)} + C.$

2. Radius of convergence: $R = 1$.

3. Interval of convergence: $(-1, 1)$.

Would you like further details or examples? Here are some related questions for exploration:

1. What is the Taylor expansion for $\ln(1-x)$, and how does it differ?
2. How would the result change if we integrated $\ln(1+x^2)$ instead?
3. Can we express the series in terms of definite integrals, such as $\int_0^x \ln(1+t) \, dt$?
4. What happens if you shift the center of the power series (e.g., expand around $x=a$)?
5. How can you verify convergence at the boundary points more rigorously?

Tip: When integrating series term by term, always verify the radius of convergence after integration!