-1
$\begingroup$

The usual power series examples $\sum_{n=1}^{\infty}a_nx^n$ in the Calculus texts always satisfy the fact that "$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$ convergent". Is there an example of power series such that $\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$ divergent?

$\endgroup$
5
$\begingroup$

Sure. Take $\displaystyle\sum_{n=0}^\infty\bigl(2+(-1)^n\bigr)x^n$, for instance.

$\endgroup$
  • $\begingroup$ Is there a name for these type of equations? $\endgroup$ – Macindows Dec 26 '18 at 16:01
  • 1
    $\begingroup$ I wrote no equation. $\endgroup$ – José Carlos Santos Dec 26 '18 at 16:02
  • $\begingroup$ I'd describe the $a_{n+1}/a_n$ sequence from this series as "oscillating". $\endgroup$ – JonathanZ Dec 26 '18 at 16:20
  • $\begingroup$ How big is its radius of convergence? Btw, I ask this question because I want to find a concrete example that can not determine its convergence at any point by ratio test (which is a classic way texts always demonstrate), but we can know it has somehow a positive radius of convergence by theoretical reason. $\endgroup$ – Eric Dec 26 '18 at 17:01
  • 1
    $\begingroup$ The radius of convergence is $\sqrt{\frac13}$. $\endgroup$ – José Carlos Santos Dec 26 '18 at 17:09
1
$\begingroup$

Consider $$\sum_{n=0}^\infty n!x^n$$ for example.

$\endgroup$
  • $\begingroup$ How big is its radius of convergence? Can it be computed? $\endgroup$ – Eric Dec 26 '18 at 17:05

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.