Is this series convergent?

Posted on 2011-10-10
Medium Priority
Last Modified: 2012-06-21
I'm doing a mock paper, with answers supplied. Here is one of the questions.

Is this series convergent?

SUM OF ((2n-1)/((4n^2)+1)
n=1 to infinity

This looks to me convergent, however...

the answer is that it is divergent by the Limit Comparison Test.

The details of the answer are:

an = ((2n-1)/((4n^2)+1)
bn = 1/n

then an>0, bn>0 and

(an/bn) = (2n^2-n)/(4n^2+1) = (2-1/n)/(4+1/n^2) -> 1/2 <> 0 as n->infinity

sum of bn from n=1 to infinity is divergent and so

sum of an is divergent from 1 to infinity by the Limit Comparison Test.

Have we not shown that an/bn tends to 1/2 as n->infinity, and hence that it is convergent ??
Question by:purplesoup
LVL 12

Accepted Solution

mwochnick earned 2000 total points
ID: 36942844
since an/bn -> 1/2 an and bn converge or diverge together - so since bn is divergent so is an

Author Comment

ID: 36943051
Oh yeh - I keep forgetting 1/n is divergent


thanks for that.

Featured Post

Free Tool: Site Down Detector

Helpful to verify reports of your own downtime, or to double check a downed website you are trying to access.

One of a set of tools we are providing to everyone as a way of saying thank you for being a part of the community.

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

When we purchase storage, we typically are advertised storage of 500GB, 1TB, 2TB and so on. However, when you actually install it into your computer, your 500GB HDD will actually show up as 465GB. Why? It has to do with the way people and computers…
Aerodynamic noise is the cause of the majority of the noise produced by helicopters. The inordinate amount of noise helicopters produce is a major problem in the both a military and civilian setting. To remedy this problem the use of an aerogel coat…
This is a video describing the growing solar energy use in Utah. This is a topic that greatly interests me and so I decided to produce a video about it.
Although Jacob Bernoulli (1654-1705) has been credited as the creator of "Binomial Distribution Table", Gottfried Leibniz (1646-1716) did his dissertation on the subject in 1666; Leibniz you may recall is the co-inventor of "Calculus" and beat Isaac…
Suggested Courses

850 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question