Evaluate the indefinite integral as a power series: $\int x^7 \ln(1 + x) \, dx$

$f(x) = C + \displaystyle\sum_{n=1}^\infty \boxed{(\ ? \ )}$

What is the radius of convergence $R$?

$R = \boxed{?}$

Neetesh Kumar | December 20, 2024

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This is the solution to Math 1c
Assignment: 11.9 Question Number 10
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Step-by-step solution:

Step 1: Recall the series expansion for $\ln(1 + x)$:

The power series for $\ln(1 + x)$ is:

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

Step 2: Multiply by $x^7$:

The integrand is $x^7 \ln(1 + x)$. Multiply the power series for $\ln(1 + x)$ by $x^7$:

$x^7 \ln(1 + x) = x^7 \cdot \displaystyle\sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n}$

Simplify the powers of $x$:

$x^7 \ln(1 + x) = \displaystyle\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{n+7}}{n}$

Step 3: Integrate term by term:

The indefinite integral becomes:

$\int x^7 \ln(1 + x) \, dx = \int \displaystyle\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{n+7}}{n} \, dx$

Integrate term by term:

$\int x^{n+7} \, dx = \frac{x^{n+8}}{n+8}$

Thus, the series for the integral is:

$\int x^7 \ln(1 + x) \, dx = C + \displaystyle\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{n+8}}{n(n+8)}$

Step 4: Determine the radius of convergence:

The power series for $\ln(1 + x)$ converges for $|x| < 1$. Multiplication and integration do not change the radius of convergence. Thus:

$R = 1$

Final Answers:

1. The power series representation is:

$f(x) = C + \displaystyle\sum_{n=1}^\infty \boxed{(-1)^{n+1} \frac{x^{n+8}}{n(n+8)}}$

2. The radius of convergence is:

$R = \boxed{1}$

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