Logarithmic approximation of $\sum_0^{N-1} \frac{1}{2i + 1}$ Can anyone confirm that it's possible to approximate the sum $\sum_0^{N-1} \frac{1}{2i + 1}$ with the $\frac{\log{N}}{2}$? And why?
It's clearly visible that the sum has a logarithmic growth over i (check wolphram) but it's unclear to me how to prove it.
 A: This is a Riemann sum for the integral from 0 to $N$ of $1/(2x+1)$, so it approximates $(1/2)\ln(2N+1)$, which is approximately $(1/2)\ln N$.
A: $\newcommand{\+}{^{\dagger}}%
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\begin{align}
\sum_{i = 0}^{N - 1}{1 \over 2i + 1}&=\half\sum_{i = 0}^{N - 1}{1 \over i + 1/2}
=
\half\bracks{\Psi\pars{\half + N} - \Psi\pars{\half}}
=
\half\Psi\pars{N + \half} + \half\,\gamma + \ln\pars{2}
\end{align}
where $\Psi\pars{z}$ is the $\it\mbox{digamma function}$ and
$\gamma = 0.577215664901533\ldots$ is the $\it\mbox{Euler-Mascheroni constant}$.
$\ds{\Psi\pars{1 \over 2} = -\gamma - 2\ln\pars{2}}$
Also,
$\ds{\Psi\pars{z} \approx \ln\pars{z} - {1 \over 2z}\ \mbox{when}\ \verts{z} \gg 1}$:
\begin{align}
\sum_{i = 0}^{N - 1}{1 \over 2i + 1}
&\approx
\half\bracks{\ln\pars{N + \half} - {1 \over 2\pars{N + 1/2}}} + \half\,\gamma + \ln\pars{2}\,,
\qquad N \gg 1
\end{align}
$$\color{#0000ff}{\large%
\sum_{i = 0}^{N - 1}{1 \over 2i + 1}
\approx
\half\ln\pars{N + \half} - {1 \over 2\pars{2N + 1}} + \half\,\gamma + \ln\pars{2}\,,
\quad N \gg 1}
$$
$\large{\bf ADDENDUM:}$ Following $\tt @Matteo$ comment:

Also,
\begin{align}
\half\sum_{i = 0}^{N - 1}{1 \over i + 1/2}
&=\half\sum_{i = 0}^{N - 1}\pars{{1 \over i + 1/2} - {1 \over i + 1}}
+ \half\bracks{\sum_{i = 0}^{N - 1}{1 \over i + 1} - \ln\pars{N}} + \half\,\ln\pars{N}
\\[3mm]&={1 \over 4}\sum_{i = 0}^{N - 1}{1 \over \pars{i + 1}\pars{i + 1/2}}
+ \half\bracks{\sum_{i = 0}^{N - 1}{1 \over i + 1} - \ln\pars{N}} + \half\,\ln\pars{N}
\\[3mm]&\stackrel{N\ \gg\ 1}{\ds{\LARGE\sim}}
{1 \over 4}\,{\Psi\pars{1} - \Psi\pars{1/2} \over 1 - 1/2} + \half\,\gamma
+ \half\,\ln\pars{N}
\\[3mm]&=
\half\braces{-\gamma - \bracks{-\gamma - 2\ln\pars{2}}} + \half\,\gamma
+ \half\,\ln\pars{N}
\\[3mm]&=\color{#0000ff}{\large\half\,\ln\pars{N + \half} - {1 \over 2\pars{2N + 1}} + \half\,\gamma + \ln\pars{2}}
\\[3mm]&\phantom{=}+
\color{#ff0000}{\underbrace{\half\bracks{%
{1 \over 2N + 1} - \ln\pars{1 + {1 \over 2N}}}}
_{\sim\ {\rm O}\pars{1/N^{2}}}}
\end{align}

