nmax = 50;
Tlog[x_] := 
        Sum[(1/k) (-1)^(k - 1) (x - 1)^k, {k, 1, 
            n}];
Do[Print["n=", 
        n]; 
Plot[{Tlog[x], Log[x]}, {x, 0.001, 3}, 
        PlotRange -> {-7.5, 7.5}];
, {n, 1, 
        nmax}]