%5En-converge-or-diverge.jpeg "(-1)^n Converge Or Diverge")
Does (-1)^n Converge or Diverge?
The sequence (-1)^n is a classic example in mathematics that demonstrates the concept of divergence.
Understanding the Sequence
The sequence (-1)^n generates alternating values:
- n = 0: (-1)^0 = 1
- n = 1: (-1)^1 = -1
- n = 2: (-1)^2 = 1
- n = 3: (-1)^3 = -1
- ... and so on
The sequence oscillates between 1 and -1 indefinitely.
Convergence vs. Divergence
A sequence converges if its terms approach a specific value as n goes to infinity. Conversely, a sequence diverges if its terms don't approach a single value.
Why (-1)^n Diverges
In the case of (-1)^n, the sequence does not approach a specific value as n increases. It continues to alternate between 1 and -1 forever. Therefore, it diverges.
Visualizing Divergence
Imagine plotting the terms of the sequence on a number line. The points would jump back and forth between 1 and -1, never settling down to a single point. This visual representation reinforces the idea that the sequence diverges.
Key Takeaway
While the sequence (-1)^n exhibits a predictable pattern, it lacks the crucial element of approaching a single value. This is why it is classified as divergent.