Discussion
Which of the following power series does not have radius of convergence 1 ?
Answer: D
The series without radius 1 is \(\frac{1}{2} + \frac{z}{2^{2}} + \frac{z^{2}}{2^{3}} + \ldots\).
Its coefficients are \(a_{n} = \frac{1}{2^{n+1}}\), so \(|a_{n}|^{1/n} \to \frac{1}{2}\) and the radius is 2. It is the geometric series \(\frac{1}{2} \cdot \frac{1}{1 - z/2}\), which converges for \(|z| \lt 2\).
The others all have radius 1. \(1 - z + z^{2} - \ldots\) is \(\frac{1}{1 + z}\). \(z - \frac{z^{2}}{2} + \ldots\) is \(\log(1 + z)\), with coefficients \(\pm\frac{1}{n}\). In \(2 - 2.3z + 3.4z^{2} - \ldots\) the coefficients grow only polynomially (whether read as products \(2 \cdot 3, 3 \cdot 4, \ldots\) or as the decimals), and \(|a_{n}|^{1/n} \to 1\).
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