$$ \lim_{x\rarr1}\frac{\sum_{k=1}^{n}x^{k}-n}{x-1} $$

The given limit is:

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

### Step-by-step solution:

1. **Sum of the series**: The expression $ \sum_{k=1}^{n} x^k $ is a geometric series. The sum of the first $n$ terms of a geometric series is given by the formula:
   \[
   \sum_{k=1}^{n} x^k = \frac{x(x^n - 1)}{x - 1}.
   \]
   However, in this case, the series starts from $k=1$, so we remove the $1$ term from the sum.

2. **Rewriting the numerator**: The numerator becomes:
   \[
   \sum_{k=1}^{n} x^k - n = \frac{x(x^n - 1)}{x - 1} - n.
   \]
   Substituting this back into the limit:
   \[
   \lim_{x \to 1} \frac{\frac{x(x^n - 1)}{x - 1} - n}{x - 1}.
   \]

3. **Indeterminate form**: As $x \to 1$, both the numerator and the denominator approach 0, which suggests we might be able to apply **L'Hopital's Rule**. First, simplify the numerator.

4. **Numerator at $x = 1$**: To avoid direct substitution, consider applying L'Hopital's Rule by differentiating the numerator and the denominator.

   - Differentiate the numerator with respect to $x$.
   - Differentiate the denominator with respect to $x$.

After applying L'Hopital's Rule and simplifying, you'll find that the final result of the limit is:

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

This is the sum of the first $n$ integers.

### Questions:
1. What is the sum formula for a geometric series?
2. How does L'Hopital's Rule work for indeterminate forms?
3. How can we verify the result using Taylor series expansion?
4. What happens if the limit approaches a different value of $x$?
5. How is the result related to the sum of integers?

### Tip:
Whenever you encounter a limit that produces a $ \frac{0}{0} $ form, L'Hopital's Rule or series expansions can be effective tools for simplifying the expression.

$$ \lim_{x \to 1} \frac{\sum_{k=1}^{n} x^k - n}{x - 1} $$

Explore the limit of the expression $ \lim_{x \to 1} \frac{\sum_{k=1}^{n} x^k - n}{x - 1} $, using concepts from calculus such as L'Hopital's Rule and geometric series. This step-by-step solution covers the sum of the first n integers and key techniques for solving indeterminate forms.

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