The problem requires differentiating a given series expansion of $f$ term-by-term to find the series expansion for $f'(x)$. The function is given as:

$f(x) = \frac{x^7}{1 - x} = \sum_{n=0}^{\infty} x^{7+n}$

Step-by-Step Solution:

1. Understanding the Series: The series given is: $\sum_{n=0}^{\infty} x^{7+n} = x^7 + x^8 + x^9 + \ldots$ This is a power series representation for $\frac{x^7}{1 - x}$.

2. Differentiating Term-by-Term: We want to differentiate $f(x)$ term-by-term to find $f'(x)$: $f'(x) = \sum_{n=0}^{\infty} \frac{d}{dx}(x^{7+n})$

3. Differentiating Each Term: The derivative of $x^{7+n}$ with respect to $x$ is: $\frac{d}{dx}(x^{7+n}) = (7+n) x^{6+n}$

4. Rewriting the Series: Substitute the differentiated terms back into the series: $f'(x) = \sum_{n=0}^{\infty} (7+n) x^{6+n}$

So, the series expansion for $f'(x)$ is:

$f'(x) = \sum_{n=0}^{\infty} (7+n) x^{6+n}$

Would you like a deeper explanation or have any questions on this?

Here are some related questions:

1. How do you differentiate a power series term-by-term in general?
2. What are the conditions under which term-by-term differentiation of an infinite series is valid?
3. How does this series compare to the Taylor series of common functions like $\ln(1 - x)$?
4. Can you find the radius of convergence for the original series?
5. How would you integrate this series term-by-term?

Tip: When differentiating a power series, ensure that the series converges within the interval you're working with, as term-by-term differentiation is only valid within the radius of convergence.