$MinPrecision = 50

Term[i_] := (2/(8i+1) + 2/(8i+2) + 1/(8i+3) - 1/2/(8i+5) -1/2/(8i+6)-1/4/(8i+7))

PiSum[n_] := Sum[Term[i]/16^i, {i,0,n}]//N

PiSum[40]

Pi//N

BBPTerm[i_] := 4/(8i+1) - 2/(8i+4)  - 1/(8i+5) -1/(8i+6)

BBPSum[n_] := Sum[Term[i]/16^i, {i,0,n}]//N

BBPSum[40]