Why does the narrator believe he will not be caught after murdering the old man?

The narrator of "The Tell-Tale Heart" attempts to argue that he is perfectly sane and that he had good reasoning for killing the old man. At the start of the story, he questions why people think him crazy:

How, then, am I mad? Hearken! and observe how healthily –– how calmly I can tell you the whole story.

He then justifies his reasoning for killing the old man. He says there was no desire for money or material goods that prompted him. Instead, he was prompted to kill because of the old man's eyes:

I think it was his eye! yes, it was this! He had the eye of a vulture --a pale blue eye, with a film over it.

Throughout the short story, the narrator continues to argue that he had completely sound and justified reasons for killing the old man. In his view, the old man deserved to die because of his eyes.
After killing the man, he chops the body into pieces and buries it under the floorboards of the house. He's very meticulous as he neatly gets rid of all visible evidence of his murder:

If still you think me mad, you will think so no longer when I describe the wise precautions I took for the concealment of the body. . . . First of all I dismembered the corpse. I cut off the head and the arms and the legs.I then took up three planks from the flooring of the chamber, and deposited all between the scantlings.

(Rather than convincing readers that the narrator is sane in this scene, readers should be questioning his sanity even more. Though he cleans up all of the visible evidence, the evidence of his crime would start to smell pretty quickly.)
Even when police officers come to the door, after hearing reports of a scream, he invites them in without fear, believing that he fully removed all evidence. He thinks that he is far too smart to be caught for his crime of murder:

There was nothing to wash out –– no stain of any kind –– no blood-spot whatever. I had been too wary for that. A tub had caught all –– ha! ha!

Ultimately, the narrator does not think he will be caught because he believes that he has been extremely careful in his process of killing the old man and hiding his body. He does not believe that anyone will be able to find clues to his guilt, and he is so confident that he invites the police officers inside. He is so sure that he will not be caught that he even places his chair over the spot where he buried the old man; he sits in this chair as he talks with the police officers. He believes that his thinking is completely right-minded; he thinks himself so intelligent that his crime cannot possibly be discovered.