1. ## Question about infinite sequences and series

I'm studying the monotonic sequence theorem but can't really understand what it is used for? Does it just prove that a sequence converges but doesn't provide an answer to what it converges to? Can someone show me an example please? Also for the squeeze theorem how do you decide what sequence you use for A and C when B is between A and C? Thanks

P.S. What is the bounded sum Test and how is it applied?

2. Originally Posted by 11rdc11
I'm studying the monotonic sequence theorem but can't really understand what it is used for? Does it just prove that a sequence converges but doesn't provide an answer to what it converges to? Can someone show me an example please? Also for the squeeze theorem how do you decide what sequence you use for A and C when B is between A and C? Thanks
The bounded monotone theorem says a sequence converges. Sometimes mathematicians only care if the sequence converges or diverges. They do not care what it converges to. Sometimes it can be really hard to find what it converges to. Thus, the theorem tells us something converges without the need to find the number.

A stupid example would be $a_n = \tfrac{1}{n}$. This sequence is monotone (why?) and bounded (why?) thus it is convergent. In this case it is easy to see convergence.

Here is an application of squeeze theorem. Say we want to find the limit of $\tfrac{1}{n}\sin \tfrac{1}{n}$. We note that $-\tfrac{1}{n} \leq \tfrac{1}{n}\sin \tfrac{1}{n} \leq \tfrac{1}{n}$. The outer sequences both converges to 0 thus the inner sequence converges to 0.

3. Thanks so its monotonic because it decreasing and seq a = 1/n and seq b = 1/(n +1) so seq of a is greater than seq b and since is bounded than seq a converges. Seq a converges to 0 because as the limit of n= infinity seq a equals 0. Is this right?

Also rather than use squeeze theorem couldn't I always use hopital rule? Thanks