Determine if `sum_(n=1)^(oo) n/(n^2-2)` converges or diverges:

Both the ratio and root tests are inconclusive.

Rewrite `n/(n^2-2)` as `1/(n-2/n)` . Note that for `n>=4` `1/n<=1/(n-2/n)` .

**So by the direct comparison test the series diverges.** (The harmonic series diverges, and every term after the second term is larger than the corresponding term of the harmonic series.)

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The ratio test: We consider `lim_(n->oo) (a_(n+1))/(a_n)`

Here we have `lim_(n->oo) ((n+1)/((n+1)^2-2))/(n/(n^2-2))`

`=lim_(n->oo)(n^3+n^2-2n-2)/(n^3+2n^2-n)=1`

so the ratio test is inconclusive.

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**Further Reading**