calculus sequence help?

Question

Determine whether the sequence is increasing, decreasing, or monotonic. Is the sequence bounded?

1) an= 1/(2n+3)

2) an= n + 1/n

can you show the steps/explain..thanks

Answer 1

1) Consider the function f(x) = 1/(2x + 3). We find
f'(x) = -2/(2x + 3)^2. The derivative is always negative, so the function is decreasing monotonically. The terms of your sequence are embedded in this function, so the sequence must also be monotonically decreasing.

Because the sequence is monotonic, it must have at least one bound. Your terms are 1/5, 1/7, 1/9, ... Clearly this is bounded above by 1/5, and below by 0 (since the limit as n -> inf = 0).

2) As above, embed this in a function of a real variable to decide about monotonicity. You can also decide about boundedness by looking at the behavior of f(x) for x >=1.