Can any kind soul help me check series

Question

anonymous · Answer

attachment

anonymous · Answer

my interval of convergence (- infinity, infinity) and radius is infinity, is it correct?

kainui · Answer

Yes, it's correct, but more importantly how did you get to your answer?

anonymous · Answer

true

miracrown · Answer

yep, perfectly correct
did you use the ratio test? @ememlove

kainui · Answer

To find the radius of convergence all you're really doing is finding what values of x cause it to fail one of your convergence tests. So in this case, we're obviously going to spring for the ratio test. For that, the sum must be less than 1 to be convergent, so if there's some value of x we can plug in that causes the limit of the ratio to be 1 or greater, that will limit our radius.

When we calculate it out however, we get:$$\lim_{n \rightarrow \infty} \frac{2(x-2)}{n+3}$$ we see that the limit =0. That means no matter what we plug in for x, it converges. Its convergence doesn't even depend on x. =)

anonymous · Answer

yep! i got that exactly.. just so unusual for me to get zero so i get paranoid.. , thanks guys! xD