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

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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))`

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

so the ratio test is inconclusive.

---------------------------------------------------------------

Approved by eNotes Editorial Team