Csdrhrt

We have to prove

$$ln Big(frac{x+1}{1-x}Big)=sum_{n≥0} frac{x^{n+1}}{n+1}P_n(x)$$

using the generatrix function of Legendre polynoms.

I don't know if it is useful, but $$int_{-1}^{1}frac{1}{1-2tx+x^2}dt=frac{1}{x}ln Big(frac{x+1}{1-x}Big)$$

The Generating function for Legendre polynomials is defined as
$$G(x,t):=sum_{n=0}^infty t^n P_n(x) $$

Using harmonic functions, we can easily derive:

$$G(x,t)=frac{1}{sqrt{1-2tx+t^2}} quad quad (1)$$

( See More Here: http://www.ucl.ac.uk/~ucahdrb/MATHM242/OutlineCD2.pdf )

Taking into account $G$'s definition, one should observe that:

$$ int_0^x G(x,t) dt = sum_{n=0}^infty frac{x^{n+1}}{n+1}P_n(x) quad quad (2)$$

Also, from equation $(1)$:

$$int_0^x G(x,t)dt=int_0^x frac{1}{sqrt{1-2tx+t^2}} dt =lnleft(left|sqrt{t^2-2xt+1}+t-xright|right)Bigg|_0^x=$$
$$lnleft(left|sqrt{1-x^2}right|right)-lnleft(left|1-xright|right)=lnleft(left|sqrt{1+x}sqrt{1-x}right|right)-lnleft(left|1-xright|right)=lnleft(left|frac{sqrt{1+x}sqrt{1-x}}{1-x}right|right)$$
$$=lnleft(left|frac{sqrt{1+x}}{sqrt{1-x}}right|right)=frac12 ln left(left|frac{1+x}{1-x}right|right) quad quad (3)$$

Now, from equations $(2),(3)$:
$$sum_{n=0}^infty frac{x^{n+1}}{n+1}P_n(x)=frac12 ln left(frac{1+x}{1-x}right)$$

assuming $(1+x)(1-x)>0$.