Convergent and Divergent Sequences

Example: The sequence does not converge.

Note that

= {-1, 1, -1, 1, -1, 1, ...}. While this sequence does exhibit a definite pattern, it does not get close to any one number, i.e. it does not seem to have a limit. To prove this statement, we will use a proof by contradiction.

Suppose that it did converge to a limit L. Then, for = 1/2 there exists a positive integer N such that

| (-1) ⁿ- L | < 1/2 for all n > N

But then, for some n > N, we have the inequality

2 = | (-1) ^{n + 1} - (-1) ⁿ | = | ((-1) ^{n + 1} - L) + (L - (-1) ⁿ ) |
- | (-1) ^{n + 1} - L | + | (-1) ⁿ - L | < 1/2 + 1/2 = 1 for n > N

which is a contradiction, since it says that 2 < 1, which is not true.

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