Want a cashmere sweater? Why not fly to London and get one cheap?

That is what Harrods, London's largest department store, will urge Americans to do in a full-page advertisement it plans to run in The New York Times on Sunday. That advertisement, the first ever in the United States for the store that bills itself as the world's largest, provides an easily understood, practical illustration of what the soaring dollar can mean to American shoppers at home and overseas.

While the strong dollar hurts U.S. companies trying to sell in foreign markets by raising the prices of their products, it lowers the cost of imports to the United States. For Americans overseas, it also makes shopping there a bargain-hunter's paradise. The strong dollar is also largely responsible for the growing U.S. merchandise trade deficit, expected to reach a record $130 billion this year and soar even higher in 1985.

The after-Christmas sale at Harrods, which starts Jan. 4 and runs for three weeks, traditionally draws shoppers from all over Europe because of its bargains. Most of the stock is marked down to about half its normal cost.

But Harrods press spokesman Nicki Cartwright said the store has never before tried to attract customers from the United States.

"The state of the dollar and the pound made the cost of coming to this country more accessible to Americans," Cartwright said.

"We don't know how many will make a special trip," she added. "But we think a lot of Americans will either advance trips or put one back to coincide with our sale."

More Americans than usual are taking advantage of travel and shopping bargains brought about by the strong dollar to visit Europe this winter. The current volume of travel, even without the cut-rate fares, "was unheard of in past seasons," said Barbara Bentz of Ask Mr. Foster, a travel agency.

"Charters are continuing through the winter, which is unusual. People are just taking advantage of low off-season rates and the strong dollar," she continued.

The pound, which as recently as five years ago was worth about $2.25, dropped yesterday to $1.16. One year ago yesterday, on Dec. 27, 1983, it was worth $1.43 -- nearly 20 percent higher than today. The dollar, on the other hand, is considered by international economists to be overvalued by about 30 percent.

That sharp decline in the pound coupled with the strong dollar may create bargains large enough for Americans who buy heavily to save enough to cover the new low winter air fares that go into effect in January.

For instance, because of the strong dollar, cashmere sweaters for women will cost Americans less at the 1985 Harrods sale than at the 1984 sale, even though their price in pounds has gone up. They were priced at 41 pounds in 1984, but went up to 47 pounds this year. The dollar value, however, dropped from $59 to $55.

The same is true for designer ties for men. They will cost $11.50 at the Harrods sale next month compared with $12.80 at the sale last January. Once again, the dollar decrease comes in spite of a jump in the cost in British pounds, from 8.95 pounds sterling last January to 9.95 next month.

Fine china sets that cost $380 during last January's sale will cost $60 less next month, and crystal goblets will be about $1 cheaper.

The quoted retail price in pounds becomes even cheaper if foreign buyers take the trouble to get a rebate of the 15 percent value added tax (VAT).

In all cases, Harrods said the prices are far lower than those for the same products in American stores. Men's cashmere coats, which cost at least $500 in the United States, can be bought during the sale for $150.

"You can hardly afford not to buy or travel overseas with the dollar at these levels," said Allen Sinai, chief economist for Shearson Lehman/American Express.