Power Series

Reading

Section 2.2, 2.3

Problems

Practice problems:

1. (Page 32) 8, 9b, 10, 13, 14, 21, 23

Challenge: 12, 17, 18

Topics to know

1. Definition of \(\displaystyle\overline{\lim_{n\to\infty}} a_n = \lim_{n\to\infty}\left(\sup_{k\geq n} a_k\right)\). If \(\displaystyle\overline{\lim_{n\to\infty}} a_n = L\), then:
   • For each \(N\) and \(\epsilon > 0\) there is a \(k > N\) such that \(a_k \geq L-\epsilon\).
   • For each \(\epsilon > 0\) there is an \(N\) such that for all \(k\geq N\) we have \(a_k \leq L+\epsilon\).
2. Radius of convergence for \(\sum C_k z^k\) dependent on \(\displaystyle L =\overline{\lim}\left|C_k\right|^{1/k}\).
   • If \(L = 0\), then \(\displaystyle L =\overline{\lim}\left|C_k\right|^{1/k}|z| = 0\), so \(|C_k z^k|\leq\frac{1}{2^k}\). Series converges absolutely for all \(z\).
   • If \(L = \infty\), then \(\left|C_k\right|^{1/k}\geq \frac{1}{|z|}\) for infinitely many \(k\). Hence \(|C_kz^k|\not\to 0\). Diverges for all \(z\neq 0\).
   • If \(L = 1/R \in (0,\infty)\):
     • If \(|z| = R(1-2\delta) < R\), then \(\displaystyle\overline{\lim} \left|C_k\right|^{1/k}|z| = 1 - 2\delta\), so \(\left|C_k\right|^{1/k}|z| < 1-\delta\) for all \(k\) sufficiently large. Series converges.
     • If \(|z| > R\) then \(\displaystyle\overline{\lim} \left|C_k\right|^{1/k}|z| > 1\) so \(C_k z^k\not\to 0\).
3. Examples 1-7 (page 27)
4. Differentiability of power series:
   • If \(\sum C_n z^n\) converges on some disc \(D(0, R)\), then so does \(\sum n C_n z^{n-1}\), since \(\displaystyle\overline{\lim}\left|n C_n\right|^{1/(n-1)} = \overline{\lim}\left|C_n\right|^{1/n}\).
   • Key idea for case where \(R=\infty\): \(\displaystyle \frac{f(z+h) - f(z)}{h} - \sum_{n=0}^\infty n C_n z^{n-1} = \sum_{n=2}^\infty C_n b_n\) where \(b_n = \frac{(z+h)^n - z^n}{h} - nz^{n-1}\leq |h|\left(|z| + 1\right)^n\).
   • So \(\displaystyle \left|\frac{f(z+h) - f(z)}{h} - \sum_{n=0}^\infty n C_n z^{n-1}\right| \leq A |h|\) (because \(\sum |C_n|(|z|+1)^n\) converges).
   • Let \(h\to 0\).
   • If \(R<\infty\) the proof is technically more difficult (see page 29).
5. Power series are infinitely differentiable.
6. Power series coefficients depend on the higher order derivatives at 0.
7. Uniqueness theorem: If series is zero when evaluated at all points of a sequence \(z_n\to 0\), then series is identically zero.
   • Inductively compute \(C_n = \lim_{z\to 0}\frac{f(z)}{z^n} = \lim_{k\to \infty}\frac{f(z_k)}{z_k^n} = 0\)
8. If two series agree on a sequence that goes to \(0\), then they are identical series.