RAW: using the exact wording of the Rainbow servant class, yes.

For all conceptual purposes, Rainbow servant turns a purely arcane caster into both an arcane AND divine casting class, as the ability granted by the Prestige class states that you use your original arcane class' spell slots to cast divine spells. (Yes, I know there is a heavy debate on the nebulous definition of "Arcane spellcasting class" VS "Divine spellcasting class", but for practical reasons, an Arcane casting class is a class that allows you to cast arcane spells. A Divine casting class is a class that allows you to cast divine spells. Again, that is the practical/logical interpretation of the term, and likely has no "official" published evidence to declare either way.)

This being the case/logical assumption, it would follow that the Rainbow Servant modifies any qualifying arcane spell caster into a dual-sourced casting class. Following that logic/assumption, the wizard (or other qualifying arcane-spell capable spellcasting class) would indeed qualify for the spell-casting requirements of Mystic Theurge, and would also qualify for ten levels of double spell progression. Congratulations, you have potentially shattered any concept of balance for spellcasting in your game.

This being said, that decision ultimately rests with your Game Master, as "rule 0" can potentially override any written rule in the game, and any GM with an eye to keep a party balanced with each other would (most likely) disallow that particular combination. I do not have evidence to support the following statement, but I believe this was one of those circumstances none of the designers anticipated being a possible issue at the time of publication, and so no regulation or counterbalance was ever formulated while 3.5 was still in publication. I hope this helps.

And, always ask your DM before attempting a build you need to question the validity of. They may save you time and energy spent working out a min/max build by telling you "no" before you do all the work, rather than telling you "no" after you put all the work in.

RAW: using the exact wording of the Rainbow Servant class, yes.

For all conceptual purposes, Rainbow Servant turns a purely arcane caster into both an arcane and divine casting class, as the ability granted by the prestige class states that you use your original arcane class' spell slots to cast divine spells. (Yes, I know there is a heavy debate on the nebulous definition of "Arcane spellcasting class" VS "Divine spellcasting class", but for practical reasons, an Arcane casting class is a class that allows you to cast arcane spells. A Divine casting class is a class that allows you to cast divine spells. Again, that is the practical/logical interpretation of the term, and likely has no "official" published evidence to declare either way.)

This being the case/logical assumption, it would follow that the Rainbow Servant modifies any qualifying arcane spell caster into a dual-sourced casting class. Following that logic/assumption, the wizard (or other qualifying arcane-spell capable spellcasting class) would indeed qualify for the spell-casting requirements of Mystic Theurge, and would also qualify for ten levels of double spell progression. Congratulations, you have potentially shattered any concept of balance for spellcasting in your game.

This being said, that decision ultimately rests with your Game Master, as "rule 0" can potentially override any written rule in the game, and any GM with an eye to keep a party balanced with each other would (most likely) disallow that particular combination. I do not have evidence to support the following statement, but I believe this was one of those circumstances none of the designers anticipated being a possible issue at the time of publication, and so no regulation or counterbalance was ever formulated while 3.5 was still in publication. I hope this helps.

And, always ask your DM before attempting a build you need to question the validity of. They may save you time and energy spent working out a min/max build by telling you "no" before you do all the work, rather than telling you "no" after you put all the work in.

RAW: using the exact wording of the Rainbow servant class, yes. For all conceptual purposes, Rainbow servant turns a purely arcane caster into both an arcane AND divine casting class, as the ability granted by the Prestige class states that you use your original arcane class' spell slots to cast divine spells. (Yes, I know there is a heavy debate on the nebulous definition of "Arcane spellcasting class" VS "Divine spellcasting class", but for practical reasons, an Arcane casting class is a class that allows you to cast arcane spells. A Divine casting class is a class that allows you to cast divine spells. Again that is the practical/logical enterpretation of the term, and likely has no "official" published evidence to declare either way.) This being the case/logical assumption, it would follow that the Rainbow Servant modifies any qualifying arcane spell caster into a dual-sourced casting class. Following that logic/assumption, the wizard (or other qualifying arcane-spell capable spellcasting class) would indeed qualify for the spell-casting requirements of Mystic Theurge, and would also qualify for ten levels of double spell progression. Congratulations, you have potentially shattered any concept of balance for spellcasting in your game. This being said, that decision ultimately rests with your Game Master, as "rule 0" can potentially override any written rule in the game, and any GM with an eye to keep a party balanced with each other would (most likely) disallow that particular combination. I do not have evidence to support the following statement, but I believe this was one of those circumstances none of the designers anticipated being a possible issue at the time of publication, and so no regulation or counterbalance was ever formulated while 3.5 was still in publication. I hope this helps. And, always ask your DM before attempting a build you need to question the validity of. They may save you time and energy spent working out a min/max build by telling you "no" before you do all the work, rather than telling you "no" after you put all the work in

RAW: using the exact wording of the Rainbow servant class, yes. 

For all conceptual purposes, Rainbow servant turns a purely arcane caster into both an arcane AND divine casting class, as the ability granted by the Prestige class states that you use your original arcane class' spell slots to cast divine spells. (Yes, I know there is a heavy debate on the nebulous definition of "Arcane spellcasting class" VS "Divine spellcasting class", but for practical reasons, an Arcane casting class is a class that allows you to cast arcane spells. A Divine casting class is a class that allows you to cast divine spells. Again, that is the practical/logical interpretation of the term, and likely has no "official" published evidence to declare either way.) 

This being the case/logical assumption, it would follow that the Rainbow Servant modifies any qualifying arcane spell caster into a dual-sourced casting class. Following that logic/assumption, the wizard (or other qualifying arcane-spell capable spellcasting class) would indeed qualify for the spell-casting requirements of Mystic Theurge, and would also qualify for ten levels of double spell progression. Congratulations, you have potentially shattered any concept of balance for spellcasting in your game. 

This being said, that decision ultimately rests with your Game Master, as "rule 0" can potentially override any written rule in the game, and any GM with an eye to keep a party balanced with each other would (most likely) disallow that particular combination. I do not have evidence to support the following statement, but I believe this was one of those circumstances none of the designers anticipated being a possible issue at the time of publication, and so no regulation or counterbalance was ever formulated while 3.5 was still in publication. I hope this helps. 

And, always ask your DM before attempting a build you need to question the validity of. They may save you time and energy spent working out a min/max build by telling you "no" before you do all the work, rather than telling you "no" after you put all the work in.