To solve the limit

$\lim_{{n \to \infty}} \frac{1 - 2 + 3 - 4 + \cdots + (2n - 1) - 2n}{n},$

we need to analyze the numerator, which is a sum of alternating positive and negative terms up to $2n$.

Step-by-Step Solution

1. Rewrite the Series in Terms of Pairs: Observe that each pair of consecutive terms, $(1 - 2), (3 - 4), \ldots, ((2n-1) - 2n)$, sums to $-1$.

   Therefore, the entire sum in the numerator can be rewritten as: $(1 - 2) + (3 - 4) + \cdots + ((2n - 1) - 2n).$ Since each pair sums to $-1$ and there are $n$ pairs, the total sum of the numerator is: $-n.$

2. Rewrite the Expression: Now we can rewrite the limit as: $\lim_{{n \to \infty}} \frac{-n}{n}.$

3. Simplify the Fraction: Simplify $\frac{-n}{n}$ to get $-1$.

4. Conclusion: The limit is: $\lim_{{n \to \infty}} \frac{1 - 2 + 3 - 4 + \cdots + (2n - 1) - 2n}{n} = -1.$

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Related Questions

1. How do we handle alternating series in general?
2. What would happen if the sequence in the numerator grew faster than linearly?
3. How does this limit relate to Cesàro summation methods?
4. Can we use the sandwich theorem for similar limits?
5. How would the result change if the sequence didn't alternate in sign?

Tip: Breaking down complex series into pairs often simplifies calculations in limits involving alternating terms.