(1) If power series $\sum c_nx^n$ converges at $x=b(b\neq0)$, this series converges at $|x|<|b|$
(2) If power series $\sum c_nx^n$ diverges at $x=d(d\neq0)$, this series diverges at $|x|>|d|$
I can't understand following proof of (2):
Let $\sum c_nd^n$ diverges. If $|x| > |d|$ $\sum c_nx^n$ does not converge
Because by (1), Convergence of $\sum c_nd^n$ is determined by convergence of $\sum c_nx^n$
Therefore $\sum c_nx^n$ diverges at $|x|>|d|$
I think explanation of second sentence is not enough. What does that sentence means?
It simply a proof by contradiction. Assume to the contrary that $\sum c_n x^n$ converges when $\vert x\vert>d$. By (1) you know that for all $\vert z\vert< \vert x\vert$, $\sum c_n z^n$ converges. In particular for $z=d$, but this is a contradition to your initial assumption.