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JH
Numerade Educator

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Problem 73 Medium Difficulty

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$ a_n = \frac{1}{2n + 3} $

Answer

The sequence is bounded since $0<a_{n} \leq \frac{1}{5}$ for all $n \geq 1$

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Video Transcript

plus determine whether this sequence is increasing, decreasing or perhaps not, Mon Atomic, then will determine whether it's founded. Now let's go ahead and show that the sequence is decreasing. So here, for any and bigger than or equal to one, we'd like to show that a N plus one is less than or equal to K N. Now this is equivalent to one over to and plus one plus three less than or equal to one over two M plus three. And this is true, since the denominator on the left hand side, it's smaller. Excuse me is larger. Larger denominator means smaller fraction. So by definition A M is decreasing. That's what it means to be decreasing. It means that the successor is no larger than the previous term. Now let's go ahead and show that is bounded. A N is bounded. Many ways to show this one of them, for example, is that we can say and is always bigger than zero. But it's always less than, for example, the first term, which is one over five. We know where over five is a one equals one over five, and we know that it's decreasing. So all the other ends, or less than one over five. On the other hand, we know that Anne is bigger than zero for any end, because positive over a positive is positive. That verifies this and these together imply that I am is a bounded sequence, so to summarize, it's founded and it's decreasing. That's our final answer.