radius of converenge, interval of convergence, center, end points check?

Question

find the the following of the series

series (1-infinity) of (n+1)*(x+4)^n / (7^n)*(5n-3)

center?
radius of converenge?
interval of convergence?
do the end points converge (absolutely or conditionally?)
do the end points diverge?

I got center: -4
radius of convergence : 7
interval of convergence: -11 I am not really sure about the last two questions.
Can someone help me out on this?

Thanks,
Mark

Answer 1

since the ratio test doesn't cover the end points you have to test them individually by substituting them in the series and trying other tests that you know.
substituting x=3 in the series gives
∑ (n+1)/(5n-3)
and the limit of (n+1)/(5n-3) is 1/5 ≠ 0 so the series diverges by the test for divergence

substituting x= -11 in the series gives
∑ (-1)^n(n+1)/(5n-3)
and the limit of (-1)^n(n+1)/(5n-3) approaches both ±1/5 that is it doesn't exist so the series diverges by the test for divergence as well

therefore the end points both diverge

the end
.