Prove the following statements concerning the convergence of power series on the unit circle \(S^1 = \{z \in \mathbb{C} : |z| = 1\}\):
\(\sum_{n=1}^\infty n z^n\) does not converge at any point of \(S^1\).
\(\sum_{n=1}^\infty \frac{z^n}{n^2}\) converges at every point of \(S^1\).
\(\sum_{n=1}^\infty \frac{z^n}{n}\) converges at every point of \(S^1\) except \(z = 1\), where it diverges.
Solution
Throughout, \(z\in S^1\), so \(\abs{z}=1\).
1. (a) \(\sum_{n=1}^\infty n z^n\) diverges.
The terms satisfy \(\abs{nz^n} = n\abs{z}^n = n \to \infty\), so they do not tend to \(0\). A series whose terms do not tend to \(0\) diverges.
2. (b) \(\sum_{n=1}^\infty z^n/n^2\) converges absolutely.
The terms satisfy \(\abs{z^n/n^2} = 1/n^2\), and \(\sum_{n\ge1} 1/n^2\) converges. By comparison the series converges absolutely, hence converges.
3. (c) At \(z=1\), \(\sum_{n=1}^\infty z^n/n\) diverges.
At \(z=1\) the series is the harmonic series \(\sum_{n\ge1}1/n\), which diverges to \(+\infty\).
4. (c) If \(z\neq1\), then \(\sum_{n=1}^\infty z^n/n\) converges.
4.1. The partial sums \(B_N \coloneqq \sum_{k=1}^N z^k\), with \(B_0\coloneqq0\), satisfy \(\abs{B_N}\le M_z \coloneqq 2/\abs{1-z}\) for all \(N\ge0\).
For \(N\ge1\), the geometric sum is \(B_N = z(1-z^N)/(1-z)\), so \[\abs{B_N} = \frac{\abs{1 - z^N}}{\abs{1 - z}} \le \frac{1 + \abs{z}^N}{\abs{1 - z}} = \frac{2}{\abs{1 - z}}.\] The bound is finite because \(z\neq1\).
4.2. For \(N > M \ge 1\), with \(a_n\coloneqq 1/n\),\[\sum_{n=M}^N a_n z^n = a_N B_N - a_M B_{M-1} + \sum_{n=M}^{N-1} (a_n - a_{n+1}) B_n.\]
This is summation by parts, using \(z^n = B_n - B_{n-1}\).
4.3. For \(N > M \ge 1\),\[\left| \sum_{n=M}^N \frac{z^n}{n} \right| \le \frac{2 M_z}{M}.\]
The sequence \(a_n=1/n\) is positive and decreasing, so \(a_n - a_{n+1}\ge0\). Steps 1 and 2 give \[\left| \sum_{n=M}^N a_n z^n \right| \le M_z a_N + M_z a_M + M_z \sum_{n=M}^{N-1} (a_n - a_{n+1}) = 2 M_z a_M.\]
4.4. Q.E.D.
By step 3, the partial sums of \(\sum z^n/n\) satisfy the Cauchy criterion, so the series converges.
5. Q.E.D.
Step 1 proves (a), step 2 proves (b), and steps 3 and 4 prove (c).