Understanding the Limit Points of the Sequence (-1^n)

To understand how 1 and -1 are limit points of the sequence (-1^n), we first need to analyze the behavior of this sequence.

The Sequence (-1^n)

The sequence (-1^n) alternates between 1 and -1. Specifically:

For even n (e.g., n 0, 2, 4, ...), -1^n 1 For odd n (e.g., n 1, 3, 5, ...), -1^n -1

Therefore, the sequence can be written as:

-1^0 1 -1^1 -1 -1^2 1 -1^3 -1

Limit Points

A point x is a limit point of a sequence if for every ε > 0 there are infinitely many terms of the sequence within the interval (x - ε, x ε).

Limit Point: -1

Choose ε 0.5, so the interval is (-1.5, -0.5). Check the sequence:

The terms of the sequence that are equal to -1 occur at all odd indices n 1, 3, 5, ...

Conclusion: Since there are infinitely many terms equal to -1 in the sequence, -1 is a limit point.

Limit Point: 1

Choose ε 0.5, so the interval is (0.5, 1.5). Check the sequence:

The terms of the sequence that are equal to 1 occur at all even indices n 0, 2, 4, ...

Conclusion: Since there are infinitely many terms equal to 1 in the sequence, 1 is also a limit point.

Summary

Both -1 and 1 are limit points of the sequence (-1^n) because, for any ε > 0, you can find infinitely many terms of the sequence that are arbitrarily close to both -1 and 1. Therefore, both points satisfy the definition of limit points.

Clarifications and Common Misunderstandings

They are not, although that is the short answer. There isn't always a limit, and there can't be several different limits when there is only one limit that the sequence approaches. In this case, the sequence does not approach a single limit because it oscillates and fails to converge to any particular value.

It is possible to define gimmicky alternatives to "limit" based on acceleration methods of finding limits when limits really exist, but this doesn't mean that these are ways to "find the real answer" when there isn't a limit. This sweeping statement needs to be qualified. I am not saying that extended notions of "limit" are pointless trickery. But they do need to be distinguished from just-plain-limits, and you do need to specify which extended notion you are using.

In this case, using standard Cauchy-style definitions, neither 1 nor -1 is a limit because at any point in the sequence, there are values later in the sequence which differ from the purported "limit" by 2. That is, there are positive values of δ (namely those in the interval (0, 2)) for which there is no n_0 such that for all n > n_0, |x_n - L| δ, where L is a potential limit.