The discussion revolves around finding the limit of the sequence defined by a_n = (n^2)(1-cos(5.2/n)) as n approaches infinity. Participants explore the behavior of the cosine function and its implications for the limit.
The discussion is active, with participants providing different perspectives on the limit. Some guidance has been offered regarding rewriting the sequence and potential methods for evaluation, but no consensus has been reached on the final limit.
Participants note the indeterminate form of the limit as n approaches infinity, which raises questions about the appropriate methods for evaluation. There is also mention of the need for further clarification on the use of Taylor series and L'Hôpital's rule.
ProBasket said:Find the limit of the sequence whose terms are given by [tex]a_n = (n^2)(1-cos(5.2/n))[/tex]
well as n->inf, cos goes to 1 right? so shouldn't the limit of this sequence be 0?
shmoe said:But [tex]n^2[/tex] goes to infinity, so your limit is an indeterminate form, infinity*0. Rewrite as:
[tex]a_n=\frac{1-\cos{\frac{5.2}{n}}}{\frac{1}{n^2}}[/tex]
and use l'hopital, or you could use the taylor series for cos.