Simple question does a prestige class that adds to your arcane caster level grant you the ability to caster higher level spells: ie a 7 level wizard has fourth level access would a 5wiz 3 arcane archer have 7 level access as his arcane class level is affectively 7 wiz?
If it just adds to caster level, then no, it would have no impact on your ability to cast higher level spells. I am, however, unaware of any prestige class which strictly adds to CL. Since you used the arcane archer as an example, I would point you to their "spells per day" text, which states that at the indicated levels you gain additional spells per day, spells known (if spontaneous), and an increased effective level of spellcasting. Given that, yes, in your example the wizard 5/arcane archer 3 would have the same casting capabilities as the straight wizard 7 (barring any traits or feats to bump the 5/3's CL). Do note, however, that you do not receive the 2 free spells to your spellbook per arcane archer level (as that is an ability not tied to the wizards spells class feature).
if the class says it increases your spells per day as if increasing your level in that class, then yes.
At the indicated levels, an eldritch knight gains new spells per day as if he had also gained a level in an arcane spellcasting class he belonged to before adding the prestige class. He does not, however, gain any other benefit a character of that class would have gained, except for additional spells per day, spells known (if he is a spontaneous spellcaster), and an increased effective level of spellcasting. If a character had more than one arcane spellcasting class before becoming an eldritch knight, he must decide to which class he adds the new level for purposes of determining spells per day.
At the indicated levels, an arcane archer gains new spells per day as if he had also gained a level in an arcane spellcasting class he belonged to before adding the prestige class. He does not, however, gain other benefits a character of that class would have gained, except for additional spells per day, spells known (if he is a spontaneous spellcaster), and an increased effective level of spellcasting. If a character had more than one arcane spellcasting class before becoming an arcane archer, he must decide to which class he adds the new level for purposes of determining spells per day.
I am, however, unaware of any prestige class which strictly adds to CL.
Closest is Lantern Bearer, which adds to CL for the purpose of overcoming spell resistance only. (It grants its own spells besides)
Thank you all very much, that helps clear things up for me.