Much has been written on the whys and wherefores of doing a welcome series for new list members.  All studies show that the best time to convert a new list member to donor status is as soon as possible after the person first signs up.  The optimum, depending on which study you read, is between 5 and 30 days — but no more.
This article lays out an eminently sensible and doable recipe for a welcome series: 3 to 5 emails over a four-week period.