Do Sharpness and Smite enchantments stack damage?

With a command block I am able to provide sharpness and smite to the same weapon. Do the effects of this stack?

Yes, the damage stacks.

To test this I gave myself 4 weapons:
Sword 1: No enchantments
Sword 2: Sharpness V
Sword 3: Smite V
Sword 4: Sharpness V and Smite V

I then added an objective to track the damage I have dealt:

/scoreboard objectives add damageDealt stat.damageDealt

And summoned a skeleton with a lot of health:

/summon Skeleton ~ ~ ~ {Attributes:[{Name:generic.maxHealth,Base:3000}],Health:3000}

Against the skeleton, for each weapon, the damageDealt statistics would consistently show:

Sword 1: 70 damage

Sword 2: 100 damage

Sword 3: 195 damage

Sword 4: 225 damage

So the Sharpness V adds 30 damage (1.5 hearts), the Smite V adds 125 damage (6.25 hearts) and both together add 155 damage (7.75), which is 30 + 125.

Therefore, it appears as though extra damages simply add with each other.