Series Convergence: \sum_{n=2}^{\infty}\frac{1+n+n^2}{\sqrt{1+n^2+n^6}}

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In summary, the conversation discusses the convergence of the series \sum_{n=2}^{\infty}a_n, where a_n = (1+n+n^2)/sqrt(1+n^2+n^6). It is determined that the series is divergent based on the fact that a_n is greater than 1/n. However, it is suggested to prove this inequality rather than just guessing and a weaker result is also possible by dividing the terms by n^2 and observing the limit as n approaches infinity.
  • #1
azatkgz
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Determine whether the series [tex]\sum_{n=2}^{\infty}a_n[/tex] converges absolutely,converges conditionally or diverges.If

[tex]a_n=\frac{1+n+n^2}{\sqrt{1+n^2+n^6}}[/tex]


For several n I get [tex]a_n>\frac{1}{n}[/tex] so I decided that this series is divergent.Right?
 
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  • #2
That would do it. But did you prove that inequality? Just looking at 'several n' basically means you are guessing.
 
  • #3
In fact the result can be much weaker, we don't need to prove the inequality for all n, or even that a_n is more than 1/n, just equal. Divide the terms through by n^2 and see what the n-th term as n --> infinity is.
 

FAQ: Series Convergence: \sum_{n=2}^{\infty}\frac{1+n+n^2}{\sqrt{1+n^2+n^6}}

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