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It was clearly mentioned in Deathly Hallows that you can't curse someone under the Invisibility cloak, since it is one of the Deathly Hallows and Death has given immunity. Even if Dumbledore was the most powerful wizard alive, how did he overcome the power of the Cloak of Death himself; that, too in such a weakened state? Here is the excerpt from the 6th book Harry Potter and the Half-Blood Prince, Chapter 27, page 584... So did i actually find a mistake in the book or is there a proper explanation?

Harry hurried over to the door leading to the spiral staircase, but his hand had only just closed upon the iron ring of the door when he heard running footsteps on the other side. He looked around at Dumbledore, who gestured him to retreat. Harry backed away, drawing his wand as he did so. The door burst open and somebody erupted through it and shouted, “Expelliarmus!” Harry’s body became instantly rigid and immobile, and he felt himself fall back against the tower wall, propped like an unsteady statue, unable to move or speak. He could not understand how it had happened — Expelliarmus was not a Freezing Charm — Then, by the light of the Mark, he saw Dumbledore’s wand flying in an arc over the edge of the ramparts and understood. . . . Dumbledore had wordlessly immobilized Harry, and the second he had taken to perform the spell had cost him the chance of defending himself.

This was while Harry was under the Invisibility Cloak.

marked as duplicate by Adamant, Rand al'Thor Jun 29 '16 at 10:30

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The Cloak does not protect the wearer

The Cloak of Invisibility has many powers. For example, it is much more effective than ordinary cloaks:

“What about the Cloak, though?” said Ron slowly. “Don’t you realize he’s right? I’ve got so used to Harry’s Cloak and how good it is, I never stopped to think. I’ve never heard of one like Harry’s. It’s infallible. We’ve never been spotted under it—”

Harry Potter and the Deathly Hallows

It also does protect itself from spells:

Accio Cloak!” roared one of the Death Eaters.

Harry seized his folds, but it made no attempt to escape. The Summoning Charm had not worked on it.

Harry Potter and the Deathly Hallows

However, it simply does not protect those concealed under it from any sort of magic. It is not only Dumbledore who was able to cast a spell on Harry while he was under the Cloak. Draco Malfoy did so as well:

Petrificus Totalus!”

Without warning, Malfoy pointed his wand at Harry, who was instantly paralyzed. As though in slow motion, he toppled out of the luggage rack and fell, with an agonizing, floor-shaking crash, at Malfoy’s feet, the Invisibility Cloak trapped beneath him, his whole body revealed with his legs still curled absurdly into the cramped kneeling position. He couldn’t move a muscle; he could only gaze up at Malfoy, who smiled broadly.

Harry Potter and the Half-Blood Prince

In other words, it does not render one literally immune to dying, contrary to its portrayal in "The Tale of the Three Brothers." Then again, it might be argued that the flaw is more in others' interpretation of the tale than the text itself.

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