%2F(n%2B2%5En)-converge-or-diverge.jpeg "(n+3^n)/(n+2^n) Converge Or Diverge")
Determining the Convergence or Divergence of (n + 3^n)/(n + 2^n)
This article explores the convergence or divergence of the sequence defined by the formula:
a<sub>n</sub> = (n + 3<sup>n</sup>)/(n + 2<sup>n</sup>)
We'll analyze the sequence using the following steps:
1. Understanding the Behavior of the Terms
As n approaches infinity, the exponential terms (3<sup>n</sup> and 2<sup>n</sup>) dominate the linear terms (n). This is because exponential functions grow much faster than linear functions. Therefore, we can simplify the expression by focusing on the dominant terms:
a<sub>n</sub> ≈ (3<sup>n</sup>)/(2<sup>n</sup>) = (3/2)<sup>n</sup>
2. Analyzing the Simplified Expression
The simplified expression (3/2)<sup>n</sup> represents a geometric sequence with a common ratio of 3/2. Since the common ratio is greater than 1, the sequence grows without bound as n approaches infinity.
3. Conclusion
Based on the analysis, the original sequence (n + 3<sup>n</sup>)/(n + 2<sup>n</sup>) diverges to infinity as n approaches infinity.
Explanation
While the initial expression includes linear terms (n), their influence becomes negligible compared to the exponential terms as n becomes large. The simplified expression highlights the dominant behavior, revealing the geometric sequence that grows infinitely.
Key Points
- When analyzing sequences with both linear and exponential terms, focus on the dominant terms as n approaches infinity.
- A geometric sequence with a common ratio greater than 1 diverges to infinity.
- The initial expression, though complex, can be simplified to reveal the underlying geometric sequence that dictates the overall behavior of the sequence.