Most archmages do take apprentices and teach them the way of magic to become a wizard. But some, very few even, gift their excess powers to someone. The Archmages that make themselves patrons are those that outlive their recipient, mostly High Elves and Gnomes.

The difference between most patrons and an Archmage is their easy access, they live, they breath and are, normally, on the Material Plane.

Expanded Spell List

The Archmage lets you choose from an expanded list of spells when you learn a warlock spell. The following spells are added to the warlock spell list for you.

Archmage Expanded Spells

Spell Level	Spells
1st	shield, magic missile
2nd	acid arrow, scorching ray
3rd	fireball, glyph of warding
4th	resilient sphere, ice storm
5th	arcane hand, teleportation circle

Arcane Knowledge

Starting at 1st level, your patron bestows upon you more knowledge of the Arcane and it's uses.

You gain proficiency in the Arcana and can add your proficiency modifier twice to any checks made with this skill.

You gain two wizard cantrips of your choice, for you these count as a warlock spells.

School of your Patron

At 6th level your patron grants you knowlege of a specific area of magecraft. Pick one wizard subclass which has a school, you gain the benefits of its 2nd level feature.

Choose 1 spell of the school of magic your patron has made a connection of 1st, 2nd or 3rd level, you can cast this spell once every long rest without providing components.

Spell Slot Master

Beginning at 10th level, your patron teaches you how to get more magic out of you.

You gain an additional spell slot of your highest Pact spell slot (5th level).

Minor Spell Mastery

Starting at 14th level, your patron gives you the power to cast a spell as a cantrip. Choose a 1st-level warlock spell (including those from the expanded spell list). You can cast this spell at 1st level without expanding a spell slot. If you want to cast this spell at a higher level, you must expand a spell slot as normal.