# Power Series Minimum Interval of Convergence

Suppose that the power series, $$\sum c_n x^n$$, converges when $$x = −4$$ and diverges when $$x = 7$$. Determine whether each statement is true, false or not possible to determine.

(a) The power series converges when $$x = 10$$.

(b) The power series converges when x = 3.

(c) The power series diverges when x = 1.

(d) The power series diverges when x = 6.

I found the radius of convergence to be at least $$4$$ and I know the series is convergent at $$-4$$ so for the minimum interval of convergence it could be either $$[-4, 4]$$ or $$[-4, 4)$$. I am not sure which one to choose based on the information given.

• You don't have to choose between $[-4,4]$ and $[-4,4)$ in order to answer the question, which asks about $1,3,6,10$ but not about $4$. – Gerry Myerson Nov 7 '19 at 4:18

a) The series cannot converge when $$x=10$$.
b) and c) The radius of convergence is at least $$4$$ so it converges at $$x=3$$ and $$x=1$$.
d) The examples $$\sum (\frac x r)^{n}$$ with $$r=5$$ and $$r=6.5$$ show that the series may or may not converge at $$x=6$$.