The Behavior and Unboundedness of the Sequence x_n Σtank from k1 to n

To determine whether the sequence x_n Σk1n tan(k) is bounded, one must analyze the behavior of the individual terms tan(k) as k increases. The purpose of this article is to explore the unbounded nature of x_n and the consequences of the unbounded tangent function.

Boundedness of tan(k)

The tangent function tan(x) is well-known for its periodic behavior with a period of π. This means that tan(x π) tan(x) for any real number x. However, since k represents integer values, the behavior of tan(k) does not repeat in a straightforward manner as k increases. Instead, the function exhibits a periodic pattern but with significant variations.

Unboundedness of tan(k)

The tangent function is unbounded and has vertical asymptotes at x (2n 1)π/2 for any integer n. This means that as x approaches these points, tan(x) approaches positive or negative infinity. Consequently, there will always be integer values k for which tan(k) will take on arbitrarily large positive or negative values, as k gets close to these asymptotes.

Implications for the Sum x_n

Given the unbounded nature of tan(k), the sum x_n Σk1n tan(k) is highly susceptible to the inclusion of these extremely large values. As n increases, the influence of these terms will dominate the behavior of the sum. This results in x_n potentially diverging to positive or negative infinity.

Visualizing the Behavior

To further understand the behavior of x_n, we can visualize the first one thousand terms of the sequence using an online graphing tool like Desmos. The Desmos graph of Tan(n) (for n 1 to 1000) reveals a fascinating tangent-like behavior, as seen in the animation. Introducing the animation allows us to observe the rapid changes and the asymptotic behavior of the function more intuitively.

Conclusion

Based on the analysis, it is clear that tan(k) is unbounded and can take on arbitrarily large positive and negative values. This unbounded nature of the tangent function implies that the sequence x_n Σk1n tan(k) is not bounded. As n increases, the sequence will include infinitely many terms k such that tan(k) is extremely large, leading to x_n tending towards positive or negative infinity. Therefore, the sequence x_n is not bounded.

For a more in-depth mathematical exploration, you can use computational tools like WolframAlpha to delve into the Taylor series expansion and further properties of the tangent function.