* Dollar lower on bullish European data
* Euro at top of ranges after strong PMI surveys
* Yen gains on Ukraine tensions
* Dollar index hits lowest in more than six months (Updates prices, adds comments, changes byline, dateline, previous LONDON)
By Sam Forgione
NEW YORK, May 6 (Reuters) - The dollar slid to an eight-week low against the euro on Tuesday on bullish European data and the growing belief that the dollar was unresponsive to positive U.S. economic news, while tensions in Ukraine drove demand for the safe-haven yen.
The euro was helped by strong surveys of service-sector purchasing managers in Spain and Italy. It remains supported by flows of capital into its southern economies, where interest rates on government debt are still much higher than those in the United States.
The fall in euro zone inflation halted last month and PMI surveys of service-sector purchasing managers made for a bullish reading on the prospects of improvement in the southern economies most hurt by four years of turmoil over government debt.
"PMIs are showing that Europe is coming out of the recession that it was in," said Douglas Borthwick, managing director at Chapdelaine Foreign Exchange in New York. "The reasons for shorting the euro are no longer valid."
Data also showed the U.S. trade deficit narrowed in March as exports rebounded. While the improvement was probably not enough to help first-quarter growth, traders viewed the dollar's weak response as another sign that the U.S. currency was immune to positive U.S. developments.
"The market is taking the view that if the dollar can't rise on good news, then the path of least resistance is weakness," said Alan Ruskin, global head of G10 FX strategy at Deutsche Bank in New York.
Traders also cited the dollar's weakness in response to positive U.S. jobs data last week as another sign of its failure to rebound, as well as continued low U.S. interest rates. The U.S. dollar index, which measures the dollar against six major currencies, on Tuesday fell to its lowest in more than six months to 79.06.
Traders said persistently low U.S. interest rates, which have remained at current levels partly on the belief that the Federal Reserve will not raise rates soon, have hurt the dollar.
"People are waiting; at a minimum they'd like to see the Treasury market responding in orthodox fashion to stronger data," said Ruskin of Deutsche Bank. "We haven't seen stronger data translate into higher yields."
Ongoing clashes in Ukraine, meanwhile, drove buying in the safe-haven yen. Pro-Russia rebels shot down a Ukrainian helicopter in fierce fighting near the eastern town of Slaviansk on Monday, while Kiev moved police special forces to the port city of Odessa to stop the rebellion spreading westward.
The euro was last up 0.41 percent against the dollar at $1.39330, while the dollar was down 0.56 percent against the Japanese yen at 101.56. The dollar was also down 0.52 percent against the Swiss franc to trade at 0.8732 francs.
The dollar index was last down 0.47 percent at 79.109. (Reporting by Sam Forgione, additional reporting by Patrick Graham in London; Editing by Chizu Nomiyama)