preliminaries

Plot[Log[(1+x)/(1-x)],{x,-1,1}]

Clear[L];
Integrate[Log[(1+x)/(1-x)] Sin[n Pi x/L],x]

Fourier Series approximation

Clear[L,NumTerms];
L=.95;
NumTerms = 5;

Table[(2/(L ))NIntegrate[Log[(1+x)/(1-x)] Sin[n Pi x/L],{x,0,L}],{n,1,NumTerms}]

Table[ Sin[n Pi x/L],{n,1,NumTerms}]

Clear[partial];
partial[x_] =Table[(2/(L ))NIntegrate[Log[(1+x)/(1-x)] Sin[n Pi x/L],{x,0,L}],{n,1,NumTerms}] . Table[ Sin[n Pi x/L],{n,1,NumTerms}]

Plot[partial[x],{x,0,L}]

Show[Plot[partial[x],{x,0,L},PlotStyle->{Blue}],
Plot[Log[(1+x)/(1-x)],{x,0,L},PlotStyle->{Red}]]

Change NumTerms and re-execute

Clear[NumTerms,L,partial];
L=.95;
NumTerms = 10;
partial[x_] =Table[(2/(L ))NIntegrate[Log[(1+x)/(1-x)] Sin[n Pi x/L],{x,0,L}],{n,1,NumTerms}] . Table[ Sin[n Pi x/L],{n,1,NumTerms}]
Show[Plot[partial[x],{x,0,L},PlotStyle->{Blue}],
Plot[Log[(1+x)/(1-x)],{x,0,L},PlotStyle->{Red}]]
Show[Plot[partial[x],{x,-3L,3L},PlotStyle->{Blue}],
Plot[Log[(1+x)/(1-x)],{x,-L,L},PlotStyle->{Red}]]