what is the easiest and economical way to travel between paris and frankfurt
any suggestions in renting an apartment for about a week in the 7th district
Take the direct high-speed ICE train. Journey time is 3:49. Purchase tickets at the German Rail site: http://www.bahn.de. The earlier you book, the cheaper it will be.
In addition to Steve's post: If you're flying add the cost from downtown Paris to one of the airports and the cost from Frankfurt airport to downtown. The train is the fastest (all in all, from downtown to downtown), easiest and usually also cheapest means of transportation between the two cities.
Have you looked at a bus trip. See Eurolines.com
I agree with the train suggstions for travel between Paris & Frankfurt.
Renting in Paris. Check out vacationinparis.com for a very large selection of apts. I would, however, suggest that you rethink your choice of the 7th. We stayed there on our 1st trip and were dismayed to discover that we weren't nearly as close to many of the places we wanted to see as it "appeared" on the map. Not nearly as many cafes/bistros or nightlife as perhaps the 5th or 6th (our favorite).