(-1)^n Cos(1/n^2)

3 min read Jun 16, 2024

Exploring the Behavior of (-1)^n cos(1/n^2)

The sequence defined by (-1)^n cos(1/n^2) presents an interesting blend of oscillatory and decaying behavior. Let's delve into its characteristics:

Oscillatory Nature:

Alternating Signs: The term (-1)^n introduces alternating signs to the sequence. For even values of 'n', the term is positive, while for odd values of 'n', it's negative. This leads to an oscillating pattern in the sequence.
Cosine Function: The cos(1/n^2) component further contributes to oscillation. As 'n' increases, 1/n^2 approaches zero, and the cosine function oscillates between -1 and 1 with increasing frequency.

Decaying Behavior:

Convergence to Zero: As 'n' approaches infinity, the term 1/n^2 tends to zero. Consequently, cos(1/n^2) approaches cos(0) which is 1. However, the (-1)^n factor continues to alternate the signs, preventing convergence to a specific value. This suggests the sequence oscillates around zero with decreasing amplitude.

Visualization:

To better understand the behavior, consider plotting the first few terms of the sequence:

n = 1: (-1)^1 * cos(1) ≈ -0.54
n = 2: (-1)^2 * cos(1/4) ≈ 0.97
n = 3: (-1)^3 * cos(1/9) ≈ -0.99
n = 4: (-1)^4 * cos(1/16) ≈ 0.999
n = 5: (-1)^5 * cos(1/25) ≈ -0.9999

This visualization reinforces that the sequence oscillates between positive and negative values with decreasing amplitude, approaching zero.

Conclusion:

The sequence (-1)^n cos(1/n^2) exhibits both oscillatory and decaying characteristics. The alternating signs due to (-1)^n and the cosine function's oscillation create a dynamic pattern. The decreasing amplitude as 'n' increases results in a convergence to zero, but not a definite value, due to the alternating sign nature. This sequence provides a fascinating example of how different mathematical functions can interact to produce complex patterns.