The answer is generally no. You may be able to count your son as a dependent in the year he started his mission, and/or the year he ended his mission, if he qualifies as a full-time student for enough months before/after his mission. Think of it this way - you can only count your missionary son as a dependent if you could also have counted him if he had gone fishing instead of serving a mission.
Also, you haven't actually paid all of your son's living expenses, not from a tax perspective. The LDS church asks missionaries or their families to provide a monthly payment to the church each month the individual is serving as a missionary. Although this money is used to help offset the missionary's expenses while serving, the monthly payment is NOT considered support for your child for tax purposes; it is considered a charitable donation.
Legal justification for the assertions made above can be found in We Believe in Being Honest: Dependency Exemptions for LDS Missionaries.
