Question: Why did David and Bathsheba's child die?
About the article "Bathsheba", section "Biblical narrative".

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When David had become old and bedridden, Adonijah, his eldest surviving son and natural heir, declared himself king.
Bathsheba, David's favorite wife, and Nathan the prophet went to David and obtained his agreement that Solomon, Bathsheba's son should became king.
David gave his final instructions, to Solomon including his promise that the line of Solomon and David will inherit the throne of Judah forever, and his request that Solomon kill his oldest enemies on his behalf.
David died and was buried on Mount Zion. 
The prophet Nathan soon after confronted David about this murder, by first telling him a story of a rich man and a poor man: The rich man had many sheep, while the poor man had only one little ewe, whom he cared for greatly.
A traveler approached the rich man for food, whereby the man took the poor man's ewe and dressed it to give to the traveler.
Hearing this story, David grew angry and replied: "As surely as the Lord lives, the man who did this deserves to die!
He must pay for that lamb four times over, because he did such a thing and had no pity.
"Nathan replied, "You are the man!
This is what the Lord, the God of Israel, says: 'I anointed you king over Israel, and I delivered you from the hand of Saul.
I gave your master's house to you, and your master's wives into your arms.
I gave you the house of Israel and Judah.
"And if all this had been too little, I would have given you even more.
Why did you despise the word of the Lord by doing what is evil in His eyes?
You struck down Uriah the Hittite with the sword and took his wife to be your own.
"You killed him with the sword of the Ammonites.
Now, therefore, the sword will never depart from your house, because you despised Me and took the wife of Uriah the Hittite to be your own.'"
NIV[http://www.biblegateway.com/passage/?search=2%20Samuel%2012:5-10;&version=31;]Nathan then informs David that his child with Bathsheba must die.
Indeed, their first child dies after seven days.
David and Bathsheba later had a second son, the future King Solomon. 
Ahitophel (; "Brother of Insipidity", or "Impiety") was a counselor of King David and a man greatly renowned for his sagacity.
At the time of Absalom's revolt he deserted David (Psalm.
41:9; 55:12-14) and espoused the cause of Absalom (2 Samuel 15:12).
David sent his old friend Hushai back to Absalom, in order that he might counteract the counsel of Ahitophel (2 Sam.
15:31-37).
Ahitophel, seeing that his good advice against David had not been followed due to Hushai's influence, correctly predicted that the revolt would fail.
He then left the camp of Absalom at once.
He returned to Giloh, his native place, and after arranging his worldly affairs, hanged himself, and was buried in the sepulcher of his fathers (2 Sam.
17:1-23).
A man named Ahitophel is also mentioned in 2 Samuel 23:34, and he is said to be the father of Eliam.
Since 2 Samuel 11:3 notes that Eliam is the father of Bathsheba, some scholars suggest that the Ahitophel of 2 Samuel 15 may in fact be Bathsheba's grandfather.
Levenson and Halpern, for example, note that "the narrator is sufficiently subtle (or guileless) to have Bathsheba's grandfather... instigate the exaction of YHWH's pound of flesh," as Nathan's curse in 2 Samuel 12:11 comes to fruition. 
The meaning of the Hebrew form of the name "Bathsheba" is "daughter of the oath", "bat" meaning daughter.
The second part of the name appears in as "shua" (signifying "wealth") (compare ).
Bathsheba was the daughter of Eliam (, who is called Ammiel in ).
Her father is identified by some scholars with Eliam mentioned in as the son of Ahithophel, who is described as the Gilonite.
(See King David's Warriors.
)Bathsheba was the wife of Uriah the Hittite, and afterward of David, by whom she gave birth to Solomon, who succeeded David as king.
(United Kingdom of Israel and Judah).
The story of David's seduction of Bathsheba, told in , is omitted in Chronicles.
The story is told that David, while walking on the roof of his palace, saw Bathsheba, who was then the wife of Uriah, having a bath.
He immediately desired her and later made her pregnant.
In an effort to conceal his sin, David summoned Uriah from the army (with whom he was on campaign) in the hope that Uriah would re-consummate his marriage and think that the child was his.
Uriah was unwilling to violate the ancient kingdom rule applying to warriors in active service.
Rather than go home to his own bed, he preferred to remain with the palace troops.
After repeated efforts to convince Uriah to have sex with Bathsheba, the king gave the order to his general, Joab, that Uriah should be placed in the front lines of the battle, where it was the most dangerous, and left to the hands of the enemy (where he was more likely to die).
David had Uriah himself carry the message that ordered his death.
After Uriah was dead, David made the now widowed Bathsheba his wife.
David's action was displeasing to the Lord, who accordingly sent Nathan the prophet to reprove the king.
After relating the parable of the rich man who took away the one little ewe lamb of his poor neighbor (II Samuel 12:1-6), and exciting the king's anger against the unrighteous act, the prophet applied the case directly to David's action with regard to Bathsheba.
The king at once confessed his sin and expressed sincere repentance.
Bathsheba's child by David was struck with a severe illness and died a few days after birth, which the king accepted as his punishment.
Nathan also noted that David's house would be cursed with turmoil because of this murder.
This came to pass years later when one of David's much-loved sons, Absalom, led an insurrection that plunged the kingdom into civil war.
Moreover, to manifest his claim to be the new king, Absalom had sexual intercourse in public with ten of his father's concubines, which could be considered a direct, tenfold divine retribution for David's taking the woman of another man.
(II Samuel 16:20–23) In David's old age, Bathsheba secured the succession to the throne of her son Solomon, instead of David's eldest surviving son Adonijah.
Some commentators have written that Bathsheba may have written at least part of Proverbs 31 as suggested by the postulated connections between King Lemuel aka King Solomon.
The story of David's adultery sets up the context for the penitential Psalm 51 (50), also known as "Miserere" ("Have mercy on me, O God").