$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} \sum_{i = 1}^{n}{1 \over 2i - 1} & = \sum_{i = 0}^{n - 1}{1 \over 2i + 1} = {1 \over 2}\sum_{i = 0}^{\infty} \pars{{1 \over i + 1/2} - {1 \over i + 1/2 + n}} \\[5mm] &= {1 \over 2}\bracks{\Psi\pars{n + {1 \over 2}} - \Psi\pars{1 \over 2}} \\[5mm] & = {1 \over 2}\ \underbrace{\braces{\Psi\pars{\bracks{\color{red}{n - {1 \over 2}}} + 1} + \gamma}}_{\ds{H_{n - 1/2}}}\ -\ {1 \over 2}\ \underbrace{\bracks{\Psi\pars{\color{red}{1 \over 2}} + \gamma}} _{\ds{\int_{0}^{1}{1 - t^{\color{red}{1/2} - 1} \over 1 - t}\,\dd t}} \\[5mm] & = H_{n - 1/2} - {1 \over 2}\int_{0}^{1}{1 - t^{- 1} \over 1 - t^{2}}\, 2t\,\dd t = {1 \over 2}\,H_{n - 1/2} + \int_{0}^{1}{\dd t \over 1 + t} \\[5mm] & = \bbox[10px,#ffd,border:1px groove navy]{{1 \over 2}\,H_{n - 1/2} + \ln\pars{2}} \end{align}\begin{align} \sum_{i = 1}^{n}{1 \over 2i - 1} & = \sum_{i = 0}^{n - 1}{1 \over 2i + 1} = {1 \over 2}\sum_{i = 0}^{\infty} \pars{{1 \over i + 1/2} - {1 \over i + 1/2 + n}} \\[5mm] &= {1 \over 2}\bracks{\Psi\pars{n + {1 \over 2}} - \Psi\pars{1 \over 2}} \\[5mm] & = {1 \over 2}\ \underbrace{\braces{\Psi\pars{\bracks{\color{red}{n - {1 \over 2}}} + 1} + \gamma}}_{\ds{H_{n - 1/2}}}\ -\ {1 \over 2}\ \underbrace{\bracks{\Psi\pars{\color{red}{1 \over 2}} + \gamma}} _{\ds{\int_{0}^{1}{1 - t^{\color{red}{1/2} - 1} \over 1 - t}\,\dd t}} \\[5mm] & = {1 \over 2}\,H_{n - 1/2} - {1 \over 2}\int_{0}^{1}{1 - t^{- 1} \over 1 - t^{2}}\, 2t\,\dd t = {1 \over 2}\,H_{n - 1/2} + \int_{0}^{1}{\dd t \over 1 + t} \\[5mm] & = \bbox[10px,#ffd,border:1px groove navy]{{1 \over 2}\,H_{n - 1/2} + \ln\pars{2}} \end{align}
