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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.

Differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f. Given f(x) = x^7 / (1 - x) = ∑ x^(7+n), find f'(x).

This problem involves finding the derivative of a power series expansion for f(x) = x^7 / (1 - x) using term-by-term differentiation. The solution provides detailed steps to compute f'(x) as ∑ (7+n)x^(6+n). Ideal for high school students studying calculus and series expansions.

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