The story of how David was anointed by Samuel to be the king of Israel can be found at 1 Samuel 17: 1- 18:5
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Samuel grieved that the Lord rejected Saul as the king
LORD said to Samuel: ”I’m sending you to Jesse in Bethlehem because I’ve selected one of his sons to be king.”
Samuel did what the Lord told him. When he went to Bethlehem to make a sacrifice, he invited Jesse and his sons to the sacrifice. When they came, he saw Eliab and thought that he will be the chosen king. LORD said: “Humans look at outward appearances, but the Lord looks into the heart”
Then Jesse called Abinadab and brought him to Samuel. LORD did not chose him as king either.
Then Jesse had Shammah come to Samuel. “The Lord has not chosen this one either,” Samuel said
Jesse made seven of his sons to pass before Samuel. And Samuel said unto Jesse, LORD has not chosen these.
Then Samuel said to Jesse, “Are all your sons here?”
Jesse said, “There remains yet the youngest, but behold, he is keeping the sheep”. Samuel said to Jesse, “Send and get him”.
And he sent and brought David in. Now David was ruddy and had beautiful eyes and was handsome.
The Lord said to Samuel, “Get up and anoint him. He is the one.”
Samuel took the horn with the oil in it, and poured the special oil on Jesse’s youngest son in front of his brothers.
1 Samuel 17: 1- 18:5