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The given mathematical expression appears to involve a limit calculation as \( n \to 1 \). The expression is:

\[
\lim_{n \to 1} \frac{\sum_{k=1}^{n} (x^k + 1) - 2n}{n - 1}
\]

To solve this, we need to break it down:

1. **Summation term**: \( \sum_{k=1}^{n} (x^k + 1) \) means we sum the expression \( x^k + 1 \) from \( k = 1 \) to \( k = n \).
   - The summation for \( x^k \) is a geometric series: 
     \[
     \sum_{k=1}^{n} x^k = \frac{x(x^n - 1)}{x - 1} \text{ if } x \neq 1
     \]
   - The sum of the constant term \( 1 \) from \( k=1 \) to \( n \) is simply \( n \).

   Thus:
   \[
   \sum_{k=1}^{n} (x^k + 1) = \frac{x(x^n - 1)}{x - 1} + n
   \]

2. **Simplification of numerator**:
   The numerator becomes:
   \[
   \frac{x(x^n - 1)}{x - 1} + n - 2n = \frac{x(x^n - 1)}{x - 1} - n
   \]

3. **Division by \( n - 1 \)**:
   Now we need to divide this expression by \( n - 1 \):
   \[
   \frac{\frac{x(x^n - 1)}{x - 1} - n}{n - 1}
   \]

4. **Taking the limit as \( n \to 1 \)**:
   To compute the limit, we might need to apply L'Hôpital's Rule because the expression takes the indeterminate form \( \frac{0}{0} \) when \( n = 1 \). Differentiating the numerator and denominator with respect to \( n \) would help.

Would you like to proceed with L'Hôpital's Rule, or shall I simplify it another way? 

Here are five related questions to deepen your understanding:

1. How does L'Hôpital's Rule apply when limits take an indeterminate form?
2. What is the formula for the sum of a geometric series, and how can it be applied in different problems?
3. How does the behavior of a function change near singular points like \( n = 1 \)?
4. What other techniques besides L'Hôpital's Rule are helpful for solving limits?
5. What role does the summation notation play in solving limits involving series?

**Tip**: Whenever you encounter indeterminate forms like \( 0/0 \) in limits, L'Hôpital's Rule is often a good method to apply.

Evaluate the limit expression: lim (n → 1) [ (∑(x^k + 1) - 2n) / (n - 1) ] where the summation is from k = 1 to n.

Explore how to solve the limit expression involving summation and series: lim (n → 1) [(∑(x^k + 1) - 2n) / (n - 1)]. This solution covers summation formulas, geometric series, and techniques such as L'Hôpital's Rule to handle indeterminate forms.

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