I am having difficulty finding radius of convergence for following function:


I tried using ratio test as shown below:

$$ \begin{align} \lim_{n\to\infty}\left| \cfrac{\cfrac{{(4x-5)}^{2(n+1)+1}}{(n+1)^{^3/_2}}}{\cfrac{{(4x-5)}^{2n+1}}{n^{^3/_2}}} \right| &= \lim_{n\to\infty}{\left| \left| (4x-5)^2 \right| \cdot \cfrac{n^{^3/_2}}{(n+1)^{^3/_2}} \right| } \\ &= \lim_{n\to\infty}{\left| \left| (4x-5)^2 \right| \cdot \cfrac{\frac{3}{2}n^{^1/_2}}{\frac{3}{2}(n+1)^{^1/_2}} \right| } \quad (\text{using l'hopital's rule}) \end{align} $$

Now I realised that this is getting me nowhere as further applications of l'hopital's rule will simply not reduce the powers of $ n $ and $ n+1 $ any further. Could someone please advise me on how to solve this question?


For a limit like $$ \lim_{n\to\infty}\frac{n^{1/2}}{(n+1)^{1/2}} $$ rewrite it as $$ \lim_{n\to\infty}\left(\frac{n}{n+1}\right)^{1/2} $$

  • $\begingroup$ Thanks for your help! I realise how simple this question is now! $\endgroup$ – LanceHAOH Sep 23 '16 at 18:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.