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# Is this series convergent?

Posted on 2011-10-10

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 ??