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.

  • 1
    $\begingroup$ 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 $4$. $\endgroup$ – 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 $x=6$.


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