Suppose the series $\sum_{n=1}^\infty a_nx^n$ converges for x =-3 and diverges for x =5
a) What can you conclude about the radius of convergence of the series?
b) Does $\sum_{n=1}^\infty a_n(-6)^n $ converge? Does $\sum_{n=1}^\infty na_n2^n $ converge?
What do you think?

Question

Suppose the series $\sum_{n=1}^\infty a_nx^n$ converges for x =-3 and diverges for x =5
a) What can you conclude about the radius of convergence of the series?
b) Does $\sum_{n=1}^\infty a_n(-6)^n $ converge? Does $\sum_{n=1}^\infty na_n2^n $ converge?
What do you think?

loser66 · Answer

@SithsAndGiggles  I know you are not online, just in case.

loser66 · Answer

I've just finish my final test. This is the problem I got stuck.

anonymous · Answer

By the ratio test, the series would converge if
$$|x|\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|<1$$
Since this is true for $x=-3$, this means we have a minimum radius of convergence of $3$. Also, given that the series diverges for $x=5$, we know that the maximum radius must be $5$.

loser66 · Answer

We are not allowed to take ratio test.:(

loser66 · Answer

However, it is not <1

anonymous · Answer

The ratio test isn't the important part here. The given series is centered about $x=0$, right? The fact that it converges for $x=-3$ is enough to know that the radius of convergence must be at least $3$.

loser66 · Answer

This is what I tried:
at x =-3, the sum is $\sum_{n\rightarrow \infty} a_n (-3)^n$ converges, that shows $lim_{n\rightarrow \infty}a_n = 0$

loser66 · Answer

not, lim of the whole thing \|dw:1430434021710:dw|