radius of conergence

[kid]

If the series an (n=0 to infinity) is conditionally convergent, the radius of convergence of the series ((n+1)anx^(2n))/6^n (n=0 to infinity)?
do you know what's the basic idea of sovling it? thanks

 

[tkhunny]

Generally, a good first guess would be the Ratio Test.

 

[kid]

yes, but in the second series using the d'Alembert rule i have a(n+1)/an which i only know that is <-1 from the first series

 

[HallsofIvy]

If the series an (n=0 to infinity), the radius of convergence of the series ((n+1)anx^(2n))/6^n (n=0 to infinity)?
do you know what's the basic idea of sovling it? thanks

Using the ratio test, as tkhunny suggests, I get \(\displaystyle \frac{(n+2)a_{n+1}x^{2n+2}}{6^{n+1}}\frac{6^n}{a_nx^{2n}}\)\(\displaystyle = 4(36)\frac{a_{n+1}}{a_n}x^2\le 1\)
so the radius of convergende depends upon \(\displaystyle a_n\).

You say "If the series an (n=0 to infinity),". Isn'r there something missing there?

 

[kid]

yeah, an is conditionally convergent (how did i miss that )...