Review : Power Series — Question 10

Question 10 – Solution

Strategy. Convergence at a point controls smaller distances absolutely; divergence controls larger distances. Boundary tests cannot be transferred automatically to the opposite side.

Step 1: Bound the radius sharply. The convergent point is distance $3$ from the center, so $R\ge 3$. The divergent point is distance $4$, so $R\le 4$: if $R>4$ that point would be inside the region of absolute convergence. Thus $\boxed{3\le R\le 4}$. Convergence at distance $3$ could be boundary convergence, and divergence at distance $4$ could be boundary divergence.

Step 2: Record only forced conclusions. $$\begin{array}{c|c|l} x&|x-2|&\text{conclusion}\\\hline 0&2&\text{absolute convergence}\\ -1&3&\text{not determined}\\ 11/2&7/2&\text{not determined}\\ 6&4&\text{not determined}\\ 7&5&\text{divergence} \end{array}$$ The absolute convergence at distance $2$ follows from the convergent point at distance $3$. The opposite point at distance $3$ need not share an endpoint test, and distances between $3$ and $4$ are not settled by the bounds.

Step 3: Construct examples proving the uncertainty. Take, starting at $n=1$ and setting $a_0=0$, $$U(x)=\sum_{n\ge 1}\frac{(-1)^n(x-2)^n}{n3^n},\qquad V(x)=\sum_{n\ge 1}\frac{(-1)^n(x-2)^n}{n4^n}.$$ For $U$, $x=5$ gives alternating harmonic convergence and $x=-2$ fails the term test; $R_U=3$. At $x=-1$ it is positive harmonic and diverges, and at $11/2$ and $6$ its terms do not tend to zero. For $V$, $x=5$ converges absolutely, while $x=-2$ is positive harmonic and diverges; $R_V=4$. At $-1$ and $11/2$ it converges absolutely, and at $6$ it converges conditionally. These two examples resolve every “not determined” entry without changing the given observations.

Step 4: Realize the entire radius range. The examples above realize $R=3$ and $R=4$. For any $3<\rho<4$, use $\boxed{\sum_{n=0}^{\infty}((x-2)/\rho)^n}$. Its radius is $\rho$; at $5$ its ratio has magnitude $3/\rho<1$, and at $-2$ it has magnitude $4/\rho>1$. Thus every radius in $[3,4]$ occurs, and no narrower range follows from the two observations alone.