$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}% \newcommand{\dd}{{\rm d}}% \newcommand{\down}{\downarrow}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\fermi}{\,{\rm f}}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\half}{{1 \over 2}}% \newcommand{\ic}{{\rm i}}% \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\ol}[1]{\overline{#1}}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ \begin{align} \sum_{\ell = 0}^{\infty}h^{\ell}\int_{-1}^{0}{\rm P}_{\ell}\pars{x} &=\int_{-1}^{0}{\dd x \over \root{1 - 2xh + h^{2}}} =\left.{\root{1 - 2xh + h^{2}} \over -h}\right\vert_{x\ =\ 0}^{x\ =\ 1} ={1 \over h}\bracks{\root{1 + h^{2}} - 1 + h} \\[3mm]&={1 \over h}\bracks{\sum_{\ell = 0}^{\infty} {1/2 \choose \ell}h^{2\ell} - 1 + h}= 1 + \sum_{\ell = 1}^{\infty} {1/2 \choose \ell}h^{2\ell - 1} \end{align}
\begin{align} {1/2 \choose \ell} &= {\Gamma\pars{3/2} \over \ell!\,\Gamma\pars{3/2 - \ell}} = {\root{\pi} \over 2}\,{1 \over \ell!\,\Gamma\pars{3/2 - \ell}} \\[3mm]&= {\root{\pi} \over 2}\, {1 \over \ell!\braces{\pi/\bracks{\Gamma\pars{\ell - 1/2}} \sin\pars{\pi\bracks{\ell - 1/2}}}} = {1 \over 2\root{\pi}}\, {\pars{-1}^{\ell + 1}\Gamma\pars{\ell - 1/2} \over \ell!} \\[2mm]&= {1 \over \root{\pi}}\, {\pars{-1}^{\ell + 1}\Gamma\pars{\ell + 1/2} \over \pars{2\ell - 1}\ell!} = {1 \over \root{\pi}}\, {\pars{-1}^{\ell + 1} \over \pars{2\ell - 1}\ell!}\, {\Gamma\pars{2\ell}\root{\pi} \over 2^{2\ell - 1}\Gamma\pars{\ell}} \\[3mm]&= {\pars{-1}^{\ell + 1} \over \pars{2\ell - 1}\ell!}\, {\pars{2\ell - 1}! \over 2^{2\ell - 1}\pars{\ell - 1}!} \end{align}
$\color{#00f}{\large\left\lbrace% \begin{array}{rcl} \int_{-1}^{0}{\rm P}_{0}\pars{x}\,\dd x & = & 1 \\[2mm] \int_{-1}^{0}{\rm P}_{2\ell - 1}\pars{x}\,\dd x & = & \pars{-1}^{\ell + 1}\, {2^{-\pars{2\ell - 1}} \over 2\ell - 1}\,{2\ell - 1 \choose \ell}\,,\quad \ell = 1,2,3,\ldots \\[2mm] && 0\ \mbox{otherwise} \end{array}\right.} $