Part 3/5:
[ a_n > a_{n+1} ]
This observation indicates that the sequence is decreasing, thereby affirming its classification as monotonic.
Example 2: Another Decreasing Sequence
The second example considers the sequence:
[ a_n = \frac{2n}{n^2 + 1} ]
Solution 1: We can show that this sequence is decreasing by confirming that:
[ a_{n+1} < a_n ]
This can be achieved using cross-multiplication of the respective terms. After simplification, we can arrive at the inequality ( 1 < n^2 + n ), which holds true for all positive integers ( n ). Thus, ( a_n ) is indeed a decreasing sequence.
Solution 2: Alternatively, we could use calculus by considering the function:
[ f(x) = \frac{2x}{x^2 + 1} ]