Consider the regrouping: 1+(−1+1)+(−1+1)+⋯=1+0+0+⋯=11 + (-1 + 1) + (-1 + 1) + \cdots = 1 + 0 + 0 + \cdots = 11+(−1+1)+(−1+1)+⋯=1+0+0+⋯=1. However, ∑n=1∞(−1)n−1=1−1+1−1+⋯\sum_{n=1}^{\infty} (-1)^{n-1} = 1 - 1 + 1 - 1 + \cdots∑n=1∞(−1)n−1=1−1+1−1+⋯ diverges. What does this illustrate?
Convergent series can always be regrouped arbitrarily without changing the sum
Regrouping divergent series may create the appearance of convergence, but only truly convergent series have a well-defined sum
The Associative Property always preserves convergence for infinite series
Divergent series never have meaningful partial sums