$MinPrecision = 2000

Sum[(Factorial[
                  2*i] / (Factorial[i] * 
                    Factorial[i]))^3 * ((42*i  +  5) / 
              2^(12*i  +  4)),  {i,  0,  2000}] // N

1/Pi // N

Sum[(Sqrt[8] / 
              9801) * ((Factorial[
                    4*i] * (1103  +  
                      26390*i))) / ((Factorial[i]^4 * 
                  396^(4*i))),  {i,  0,  500}] // N