Table[{2^i,N[Sum[1/(Prime[n]*Prime[n+1]*Prime[n+2]*Prime[n+3]),{n,1,2^i}],32]},{i,1,16}]

{
{2,    0.0056277056277056277056277056277056},
{4,    0.0058862705921529450941215647098000},
{8,    0.0059254622594394833512466346903019},
{16,   0.0059288990457000983261002101022411},
{32,   0.0059291820859389597918672235999557},
{64,   0.0059292021666453483208922629720436},
{128,  0.0059292036204543131908628561182036},
{256,  0.0059292037316926009949991935234589},
{512,  0.0059292037404142987122041319598083},
{1024, 0.0059292037411235282269227656294648},
{2048, 0.0059292037411832932853761242415555},
{4096, 0.0059292037411885209554219846722601},
{8192, 0.0059292037411889920122927264466581},
{16384,0.0059292037411890355339271171606028},
{32768,0.0059292037411890396453738185905574},
{65536,0.0059292037411890400411611078652457}}

The PNT guarantees it converges: summands are O(1/(n^4 log(n^4))). So the error from finite truncation will be less than 1/n^3. Which is pretty much what is shown in the response from Bill Nelson.