Exercise A.1-2

Show that \(\sum_{k = 1}^n 1/(2k - 1) = \ln(\sqrt n) + O(1)\) by manipulating the harmonic series.

\[\begin {align} \sum_{k = 1}^n \frac 1 {(2k - 1)} & = 1 + \frac 1 3 + \frac 1 5 + \cdots + \frac 1 {2n - 1} \\ & = \left(1 + \frac 1 2 + \frac 1 3 + \cdots + \frac 1 {2n} \right) - \frac 1 2 \left(1 + \frac 1 2 + \frac 1 3 + \cdots + \frac 1 n \right)\\ & = \sum_{k = 1}^{2n} \frac 1 k - \frac 1 2 \sum_{k = 1}^n \frac 1 k \\ & = \ln 2n + O(1) - \frac 1 2 \left(\ln n + O(1)\right) \\ & = \ln 2 + \ln n + O(1) - \frac 1 2 \left(\ln n + O(1)\right) \\ & = \ln n - \frac 1 2 \ln n + \ln 2 + O(1) - \frac 1 2 O(1) \\ & = \frac 1 2 \ln n + O(1) \\ & = \ln \sqrt n + O(1) \end {align}\]