1. Finding examples of sequences.

I am currently in an intro to real analysis class, and we are on limits now. The last time I had to deal with limits was in my last calculus class two years ago. So I am very rusty when it comes to limits. Any help, hints, suggestions, and/or corrections is greatly appreciated. Thank you for your time!

(A) Give an example of a bounded sequence that diverges.

* In class we already had the example of the sequence $(-1)^n$ . Since it keeps bouncing back and forth.
I was thinking maybe a sequence dealing with a trig function say sin x or cos x maybe?

I'm stuck on B and C.

(B) Give an example of a sequence of positive real numbers, { $a_{n}$}, where { $a_{n}$} converges and .

(C) Give an example of a sequence of positive real numbers, { $a_{n}$}, such that but { $a_{n}$} diverges.

2. b) $a_n = \frac{1}{n}$

c) $a_n = n$

3. Originally Posted by Plato
b) $a_n = \frac{1}{n}$
Thanks, Plato!
So I wrote out terms in this sequence to get {1, 1/2, 1/3, 1/4, ...}
So if I did $a_{n+1}$/ $a_{n}$, I get the terms:
{1/2, 2/3, 3/4, ...} which I can see eventually converges to one since the numerator is always one less than the denominator. However, how can I show algebraically that the limit = 1? And also how can I show algebraically or with theorems that the series {1/n} converges? It's harmonic, right?

Originally Posted by Plato
c) $a_n = n$
Terms in this sequence: {1,2,3,...}
So if I did $a_{n+1}$/ $a_{n}$, I get:
{2,3/2,4/3,5/4,...} which I can 'see' diverges since the numerator is always one more than the denominator. However, how can I show algebraically that the limit is still = 1? And also how can I show algebraically or with theorems that the series {1/n} diverges?

Thanks again!

4. When I provided the examples, I assumed that you knew basic limits of sequences.
Part (b) for this:
$a_n = \frac{1}{n} \Rightarrow \quad \left( {a_n } \right) \to 0\,\& \,\left( {\frac{{a_{n + 1} }}{{a_n }}} \right) = \left( {\frac{n}{{n + 1}}} \right) \to 1$.

Part (c) is:
$b_n = n \Rightarrow \quad \left( {b_n } \right) \to \infty \,\& \,\left( {\frac{{b_{n + 1} }}
{{b_n }}} \right) = \left( {\frac{{n + 1}}
{n}} \right) \to 1$
.

5. Originally Posted by Plato
When I provided the examples, I assumed that you knew basic limits of sequences.
Thank you very much for your patience and clarifications.
I understand now. It took a while for me to refresh what I'd learned from calculus days.

As for a bounded sequence that diverges, my example is {cos (n $\pi$)}. Since this sequence is bounded by [-1, 1] but the terms alternate between -1 and 1 therefore the sequence diverges.

Is my explanation correct?