%5En%2Fn-limit.jpeg "(-1)^n/n Limit")
Exploring the Limit of (-1)^n/n as n Approaches Infinity
In the realm of calculus, understanding limits is crucial. One interesting limit to analyze is that of (-1)^n/n as n approaches infinity. This seemingly simple sequence holds intriguing behavior and highlights key concepts in limit theory.
Understanding the Sequence
The sequence (-1)^n/n alternates between positive and negative values depending on the value of n. For even values of n, the term is positive, while for odd values, it's negative. Moreover, the absolute value of each term decreases as n increases.
Analyzing the Limit
Let's delve into the behavior of the sequence as n approaches infinity:
- Oscillation: Due to the alternating nature of (-1)^n, the sequence oscillates between positive and negative values. This prevents the sequence from converging to a single value.
- Convergence to Zero: While the sequence oscillates, the absolute value of each term gets progressively smaller. This suggests that the sequence might be approaching zero in the limit.
Formal Proof
To formally prove that the limit of (-1)^n/n as n approaches infinity is 0, we can use the Squeeze Theorem.
- Bound the Sequence: We know that -1/n ≤ (-1)^n/n ≤ 1/n for all n ≥ 1.
- Limits of Bounds: The limits of both -1/n and 1/n as n approaches infinity are 0.
- Squeeze Theorem: Since the sequence (-1)^n/n is bounded between two sequences that both converge to 0, the Squeeze Theorem implies that the limit of (-1)^n/n as n approaches infinity is also 0.
Conclusion
Therefore, even though the sequence (-1)^n/n oscillates, its limit as n approaches infinity is 0. This demonstrates that a sequence can approach a limit even if it doesn't converge to a specific value. The Squeeze Theorem proves to be a powerful tool for analyzing the behavior of such sequences. The concept of oscillation and convergence to zero in this limit provides valuable insights into the nature of sequences and their limiting behavior.