What evidence suggests that Duncan is a weak king in Shakespeare's Macbeth?

King Duncan is a kind and gracious king, but he has definable weaknesses. Throughout the first act he exhibits a susceptibility to being challenged and overcome, since he is unable to quell rebellions himself and must rely upon his warriors. Further in the play, he fails to read enmity in the faces of his foes, and he misjudges the natures of others and the dangers of certain situations. Sadly, he cannot even protect his own sons.

Here are examples of Duncan's weaknesses:

• Unable to quell rebellions on his own

In the first act, as the rebel Macdonwald is defeated by Macbeth, King Duncan is not near the battle as many a king would be. Instead, he is in a camp and awaits word of the battle. His son Malcolm tells Duncan that an officer approaches who fought against his captivity, and Malcolm asks the soldier to provide the king with a report of the battle. So, the officer relates the brutal slaying of the rebel Macdonwald by the brave Macbeth. As he listens, the king seems an observer, and certainly not a participant, of the battle; and then he praises Macbeth, "O valiant cousin! Worthy gentleman!" (1.2.34)

• He displays an emotionalism that is unbecoming of a king

When Duncan announces his heir, he becomes so effusive that he can hardly say the words:

My plenteous joys,
Wanton in fullness, seek to hide themselves
In drops of sorrow. (1.4.39-41)

• He is unable to know what others are thinking, and unaware of the dangers of a setting.

As King Duncan approaches the castle of Macbeth, he is unable to sense any danger from the ruthless warrior on whom he has bestowed the title of Thane of Cawdor. Instead, he feels everything is lovely:

This castle hath a pleasant seat; the air
Nimbly and sweetly recommends itself
Unto our gentle senses (1.6.1-3)

Also unable to detect any danger from Lady Macbeth, who has just finished asking the preternatural spirits to unsex her, King Duncan addresses her with naïveté, saying that although his visit may be inconvenient, she should ask God to reward him for coming because his love prompted his visit.

See, see, our honored hostess!
The love that follows us sometime is our trouble,
Which still we thank as love. Herein I teach you
How you shall bid God 'ield us for your pains
And thank you for your trouble. (1.6.10-14)

Further, he misjudges both Macbeth and Lady Macbeth:

And his great love, sharp as his spur, hath help him

To his home before us. Fair and noble hostess,

We are your guest tonight. (1.6.23-25)

• Duncan is too meek in his position as king

In his soliloquy in Scene 7 of Act I, Macbeth reflects upon his relative, King Duncan, who is so meek and virtuous that his murder will "plead like angels" against his "horrid deed." 

...his virtues
Will plead like angels trumpet-tongued against
The deep damnation of his taking-off (1.7.18-20)

Lady Macbeth also recognizes Duncan's meekness as she tells her husband that she could have done the deed if Duncan had not reminded her of her father:

                 ...Had he not resembled
My father as he slept, I had done't (12.2.12-13)

Certainly, Shakespearean scholars, such as Janet Adelman, have alluded to King Duncan's nurturing nature, his "womanish softness," and his child-like trust and inability to interpret what is in people's faces. He is a good man, but too weak for the world in which he lives.