From the series solution of a differential equation, I obtained the following recurrence relation:
$a_{j+2}=a_j \left(\frac{(j+1)(j+3)-n(n+2)}{(j+2)(j+3)}\right),$ where $n$ is some constant. $\\$
From the ratio test we get the limit of the ratio of successive terms as 1 as j tends to infinity. How can we conclude the convergence or divergence of such a series?
The answer says that the series with the above coefficients diverges for x=1.
Thanks for the help in advance.
You are interested in the power series with these coefficients, aren't you ? In that case, as you say the ratio test tell you that the series converges for $|x|<1$ and diverges for $|x|>1$. But it tells you nothing when $|x|=1$, in particular when $x=1$. In that case, you should use some different test or comparison argument (for instance with a harmonic series).