Does this sum converge?

Questions
1

Does the below infinite sum involving primes converge?
sum(1/(Prime(n)*Prime(n+1)*Prime(n+2)*Prime(n+3)),n=1… infinity)
Free WolframAlpha app runs out of time while evaluating this sum…
The term values which free WolframAlpha app shows for this sum (starting from the term 41 and on) is 0.0059292…
If 0.0059292 is indeed the value for this sum, could then more significant digits be produced?

Thanks,
Best Regards,
Alexander R. Povolotsky

2

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}}
3

Thanks, so it appears that this sum does converge.

4

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.