determine whether the series converges or diverges, and explain which test is applicable to determine the result: `sum_(n=1)^oon/(n^2-2)`

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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.)


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))`


so the ratio test is inconclusive.


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