W[p_,q_]:=Integrate[(1-x^(1/p))^q,{x,0,1}]^(-1);

w[d_,n_]:=(∏_(k=1)^(n-1))(((((d*k+1))^2)-1)/(((d*k+1))^2));

DiscretePlot[{4((w[2.,n]*(2n))/(2n+1)),4((w[2.,n]*(2n+.5))/(2n+1)),4(∫_0^1)(((1.-(x^2)))^(1/2))\[DifferentialD]x,4w[2.,n]},{n,1,131}]

DiscretePlot[{((w[3,n]*(3n))/(3n+1)),((w[3,n]*(3n+1/2))/(3n+1)),(∫_0^1)(((1-(x^3)))^(1/3))\[DifferentialD]x,w[3,n]},{n,1,100}]

w22[n_]:=4Product[((2k+1)^2-1)/(2k+1)^2,{k,1,n-1}];
w20[n_]:=w22[n]*2n/(2n+1);
w21[n_]:=w22[n]*(2n+.5)/(2n+1);

DiscretePlot[{w20[n],w21[n],4Integrate[(1-x^2)^(1/2),{x,0,1}],w22[n]},{n,1,100}]

w32[n_]:=Product[((3k+1)^2-1)/(3k+1)^2,{k,1,n-1}];
w30[n_]:=w32[n]*3n/(3n+1);
w31[n_]:=w32[n]*(3n+.5)/(3n+1);

N[Table[{w30[100],Integrate[(1-x^3)^(1/3),{x,0,1}],w31[100],w32[100]},{n,10000,10000}],12]

w42[n_]:=Product[((4k+1)^2-1)/(4k+1)^2,{k,1,n-1}];
w40[n_]:=w42[n]*4n/(4n+1);
w41[n_]:=w42[n]*(4n+1/2)/(4n+1);

N[Table[{w40[100],Integrate[(1-x^4)^(1/4),{x,0,1}],w41[100],w42[100]},{n,10000,10000}],12]

DiscretePlot[{w30[n]},{n,100,1000}]
DiscretePlot[{w31[n]},{n,100,1000}]
DiscretePlot[{w32[n]},{n,100,1000}]
DiscretePlot[{w30[n],w31[n],Integrate[(1-x^3)^(1/3),{x,0,1}],w32[n]},{n,100,1000}]

DiscretePlot[{w40[n]},{n,100,1000}]
DiscretePlot[{w41[n]},{n,100,1000}]
DiscretePlot[{w42[n]},{n,100,1000}]
DiscretePlot[{w40[n],w41[n],Integrate[(1-x^4)^(1/4),{x,0,1}],w42[n]},{n,100,1000}]

w52[n_]:=Product[((5k+1)^2-1)/(5k+1)^2,{k,1,n-1}];
w50[n_]:=w52[n]*5n/(5n+1);
w51[n_]:=w52[n]*(5n+1/2)/(5n+1);

DiscretePlot[{w50[n],w51[n],Integrate[(1-x^5)^(1/5),{x,0,1}],w52[n]},{n,100,1000}]