I was wondering if anyone could offer any tips or hints as to how to tackle this question, I've tried looking around on here and in my notes but I can't seem to find anything.

Consider the sequence

$$ {a_n=\frac{x^n}{n^3}}. $$

B1. Apply the ratio test to find out for which values of $x$ the sequence $(a_n)$ converges.

B2. For which value(s) of $x$ does the ratio test provide no information?

B3. Does $(a_n)$ converge or diverge for these values? Justify your answers.

B4. Give an example of a sequence that converges if $-1<x<1$ and diverges otherwise. Justify your answer.

  • $\begingroup$ In B$, you ask [...] converges if $1<x<1$ [...]. Do you mean $-1<x<1$? $\endgroup$ – Alain Remillard Oct 31 '19 at 15:32
  • $\begingroup$ Ah yes I do sorry $\endgroup$ – CAHL632 Oct 31 '19 at 16:55

By ratio test we obtain

$$\left|\frac{x^{n+1}}{(n+1)^3}\frac{n^3}{x^n}\right|=|x|\left(\frac n{n+1}\right)^3 \to |x|$$

then for convergence we need $|x|<1$, then study separately the cases $|x|=1$.

  • $\begingroup$ Thank you for this, do you have any suggestions on tackling B4 or an example I could use? $\endgroup$ – CAHL632 Nov 1 '19 at 19:06
  • $\begingroup$ @CAHL632 We have that $\frac{\left(\frac12\right)^n}{n^3}$ converges and $\frac{2^n}{n^3}$ diverges. $\endgroup$ – user Nov 1 '19 at 21:04
  • $\begingroup$ I think the question is looking for just one sequence which adheres to both rules instead of an example of each. $\endgroup$ – CAHL632 Nov 2 '19 at 11:45

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