Fourier–Legendre series

Remarks

The factor $(2n+1)/2$ comes from the orthogonality relations $\int_{-1}^{1} P_m P_n \, dx = \frac{2}{2n+1} \delta_{mn}$. For $f$ in the space $L^2([-1, 1])$, the series converges to $f$ in $L^2$, because the Legendre polynomials form a complete orthogonal system there.

Example: let $f(x) = x^2$. Then $a_0 = \frac12 \int_{-1}^1 x^2 \, dx = \frac13$, $a_1 = \frac32 \int_{-1}^1 x^3 \, dx = 0$, and $a_2 = \frac52 \int_{-1}^1 x^2 \cdot \frac{3x^2 - 1}{2} \, dx = \frac52 \cdot \frac{4}{15} = \frac23$. All $a_n$ with $n \ge 3$ vanish, since $x^2$ is a combination of $P_0$ and $P_2$ and is orthogonal to $P_n$ for $n \ge 3$. So the series is $\frac13 P_0 + \frac23 P_2$, which equals $x^2$.

Non-example: $f(x) = 1/(1 - x)$ on $[-1, 1)$ is not integrable near $x = 1$. Already $a_0 = \frac12 \int_{-1}^{1} \frac{dx}{1 - x}$ diverges, so $f$ has no Fourier–Legendre series.