Solved

# Is this series convergent?

Posted on 2011-10-10
Medium Priority
219 Views
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:

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

now
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 ??
0
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
0

Author Comment

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

http://en.wikipedia.org/wiki/Convergent_series

thanks for that.
0

## Featured Post

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
Course of the Month15 days, 23 hours left to enroll