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.

  • $\begingroup$ Can you show your dif eq? $\endgroup$ – cgiovanardi Jan 30 '18 at 19:16

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).

  • $\begingroup$ That is exactly my question. Could you suggest a method to show that the series diverges for x=1? $\endgroup$ – Tejas P Jan 30 '18 at 15:41
  • $\begingroup$ Yes, I don't know the answer. I just wanted to clarify the statement. $\endgroup$ – Pablo De Napoli Jan 30 '18 at 15:43

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