2
$\begingroup$

Determine the radius of convergence, R, of the power series $\sum \limits_{n=0}^\infty a_nx^n$ where $a_n = 10^{−n} + 10^n$.
I know the series diverges and the radius of convergence is 0 but how do I prove it?

$\endgroup$
1
  • $\begingroup$ How do you know that the series diverges? How do you know that the radius of convergence is $0$? Have you tried $x=\frac{1}{100}$? $\endgroup$ Mar 28 '16 at 19:26
2
$\begingroup$

Look at the ratio test:

$$ \lim_{x\rightarrow\infty}\left|\frac{a_{n+1}x^{n+1}}{a_nx^n}\right|=\lim_{n\rightarrow\infty}\left|\frac{(10^{-n-1}+10^{n+1})x^{n+1}}{(10^n+10^{-n})x^n}\right|=\lim_{n\rightarrow\infty}\left|\frac{(10^{-n-1}+10^{n+1})x}{(10^n+10^{-n})}\right| $$

For what values of $x$ is this limit less than $1$? To make this easier, we can divide the numerator and denominator by $10^n$ to get that this limit is the same as:

$$ \lim_{n\rightarrow\infty}\frac{(10^{-2n-1}+10)|x|}{(1+10^{-2n})}. $$

As $n$ increases, the limit approaches $10|x|$, so we can find appropriate $x$'s.

$\endgroup$
2
$\begingroup$

The series does not have the radius of convergence $0,$ but rather $1/10.$

$\endgroup$

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.