First of all that is not a series. It is a sequence.
It would be a series if it had + signs between them,
like this 10+4+3+11+15+...

But assuming they meant "sequence", I'll have to ask you a question. 

Did you make a typo and the first number should have been 1, not 10?

If the first number had been 1, then we would have this pattern:

"ONE" is the smallest counting number that is spelled with 3 letters.
"FOUR" is the smallest counting number that is spelled with 4 letters.
"THREE" is the smallest counting number that is spelled with 5 letters.
"ELEVEN" is the smallest counting number that is spelled with 6 letters.
"FIFTEEN" is the smallest counting number that is spelled with 7 letters.

then for the next term, we could say:

"THIRTEEN" is the smallest counting number that is spelled with 8 letters.

Trouble is, 1 is not the first term, but 10 is.  Now, "ten" is spelled with
3 letters, just as "one" is.  However 10 ruins the pattern from the very start
because it's not the SMALLEST counting number spelled with 3 letters.

That means the next number in the sequence after 15, could be 13, 14, or 19,
since those are all spelled with 8 letters. And with the "ten" being first
instead of "one", there can be no requirement that it be the smallest integer
spelled with 8 letters.

Maybe you intended for the sequence to be 1,4,3,11,15.  If so, 13 is the
answer. Otherwise it could be 13, 14, or 19.

Now here is a formula for a sequence that has those first five terms:




Here 
     
      
        
     

Then  

That will do. But I think you may have mistyped a
10 for a 1.  If so. then the next term is 13.

Edwin