anonymous
  • anonymous
Series exercie
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
\[\sum_{1}^{\infty} \frac{ n }{ n ^{2} +1 }\]
anonymous
  • anonymous
@Owlcoffee
Owlcoffee
  • Owlcoffee
Okay, what's the objective?

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anonymous
  • anonymous
to find if it is convergent,conditonally convergent or divergent
anonymous
  • anonymous
@ayeshaafzal221 helped me yesterday with it,I just wanna see another way of solving it
anonymous
  • anonymous
Help?
mathmate
  • mathmate
Hint: use the integral test.
anonymous
  • anonymous
I got it like this: \[\lim_{n \rightarrow \infty} \frac{ \frac{ n }{ n ^{2} } }{ \frac{ n ^{2} }{ n ^{2} +1}} = \lim_{n \rightarrow \infty} \frac{ n }{ 2 } =\frac{ \infty }{ 2 }=0\]
anonymous
  • anonymous
Is my solution right?
ganeshie8
  • ganeshie8
You could also use the comparison test : for \(n\ge 1\), we have \(n^2+1 \le n^2+n^2 = 2n^2\) \(\implies \dfrac{1}{n^2+1}\ge \dfrac{1}{2n^2}\) \(\implies \dfrac{n}{n^2+1}\ge \dfrac{n}{2n^2} = \dfrac{1}{2n}\) Since \(\sum \dfrac{1}{2n}\) doesn't converge as this is a harmonic series, the series \(\sum \dfrac{n}{n^2+1}\) doesn't converge by comparison test
anonymous
  • anonymous
So my solution is not right?
ganeshie8
  • ganeshie8
I would give you 1/10 for the attempt because I don't really get what test you have used..
ganeshie8
  • ganeshie8
Also why do you think \(\dfrac{\infty}{2}\) is anything close to \(0\) ?
mathmate
  • mathmate
\(\large\frac{n}{n^2+1}\ne\frac{n/n^2}{n^2/n^2+1}\) so it is not the equivalent series, in any case \(\inf/2=\inf\)
anonymous
  • anonymous
i meant infinite,not 0,sorry
ganeshie8
  • ganeshie8
do you know what does it mean to 'converge' or 'diverge' ?
anonymous
  • anonymous
Nope,I know if it dose not equal 0,it means it is divergent,if it is 0,then it's complicated
anonymous
  • anonymous
does*
ganeshie8
  • ganeshie8
before trying the convergence tests, you need to know the meaning of terms "convergent sreies" and "divergent series"
anonymous
  • anonymous
can I use L'Hospital for this exercise?
anonymous
  • anonymous
I get 1/2 if I use L'Hospital and it means it is divergent,right?

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