Part 4/5:
Using the quotient rule to find the derivative, we simplify and establish that the derivative is negative when ( n > 1 ), thereby confirming that the function (and thus the sequence) is decreasing on the interval ( 1 ) to ( \infty ).
Implications of Monotonicity
The behavior of monotonic sequences has significant implications in mathematical analysis, particularly in understanding convergence and limits. If a sequence is bounded and monotonic (either increasing or decreasing), it is guaranteed to converge to a limit, a crucial property that forms the foundation for many results in real analysis.