Write SQL query to solve given problem: Which federal republic country in Europe has the most provinces, and what proportion of GDP is devoted to services?
Calculate the population density as well.. Keep the solution inside sql tag ```sql [SQL-Query] ```
SELECT T1.Country, T2.Service , SUM(T1.Population) / SUM(T1.Area) FROM province AS T1 INNER JOIN economy AS T2 ON T1.Country = T2.Country WHERE T1.Country IN ( SELECT Country FROM encompasses WHERE Continent = 'Europe' ) GROUP BY T1.Country, T2.Service ORDER BY COUNT(T1.Name) DESC LIMIT 1