b)
c)
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 . 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, { }, where { } converges and .
(C) Give an example of a sequence of positive real numbers, { }, such that but { } diverges.
Thanks, Plato!
So I wrote out terms in this sequence to get {1, 1/2, 1/3, 1/4, ...}
So if I did / , 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?
Terms in this sequence: {1,2,3,...}
So if I did / , 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!
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 )}. Since this sequence is bounded by [-1, 1] but the terms alternate between -1 and 1 therefore the sequence diverges.
Is my explanation correct?