f(a,b):=sum(1/((2*n-1)*(2*n+3)),n,a,b);

ratsimp(f(a,b));

approx:f(1,10000);

float(approx);

'integrate(1/((2*x-1)*(2*x+3)),x,1,inf);

integrate(1/((2*x-1)*(2*x+3)),x,1,inf);

float(%);