Depending on if the game has aged well onto modern systems, and has a good story or mechanics I'll generally enjoy it regardless of its age.
I will say that there are a bunch of series that I did get tired of as they got newer, for various reasons.
Most of Nintendo's games other than The Legend of Zelda that are series I got tired of (Starfox I enjoyed as well, but they were very sparing in releasing those and the latest one looked like it belonged on the NES, not the fucking Wii:U so it looked like crap to me especially compared to the last they released regarding StarFox). I enjoyed Paper Mario: The Thousand Year Door because it actually had good writing and gameplay, but none of the later ones interested me, in fact no Mario game after Super Mario Galaxy interested me. Smash was fun, but I really liked how they shook things up by giving Brawl a story (too bad Nintendo then took their ball and walked away afterwards because they got pissed all the cutscenes ended up on Youtube, so we're never going to see something so glorious again as long as Nintendo acts like children in regards to the internet). But yeah, I ended up selling my Wii because I just ended up playing more old Gamecube games than actual newly released Wii titles, and I got bored of them.
But as for other series I dropped since I was young: I stopped playing COD after Black Ops 1 and had skipped over a bunch up to that game already. The Big Red One was my favourite COD game and the campaign was what got me into the series.
Halo I stopped after #3, I felt that the ending was adequate and that it was a good place for me to stop while I still enjoyed it, and I'm glad I did.
Same went for Assassin's Creed IV: Black Flag, it was a great game that expanded beautifully off of the gimmick in III, but I could already get the feeling by some elements of the writing that I was starting to get tired of it, so I dropped that series while I still enjoyed it as well, haven't played any past 4.
The new Fallout looked like crap to me outside of visuals whereas I ran into some very bad bugs with Fallout 1 that made me distasteful of it.
IDK. There's something of a balance I think a lot of series tend to lose the longer they go on. Early titles IMO tend to end up with a lot of bugs or unrefined and basic mechanics, whereas later and more modern games feel incredibly copy-pasty, and don't really feel as gripping or fresh as older titles. Immediate sequels of original games shortly afterwards tend to be the best the game series will ever likely to be IMO (unless its like Deus Ex, or Dungeon Seige with Ross, where the devs hit a home run and make an amazing game on the first try, then they usually end up like movies, where it becomes extremely difficult for any sequel to live up to the expectations of the previous title without being a carbon copy) because there you now know which mechanics can be refined and how to make the game well without coming across as repetitive. Unless you can then make the game even more interesting in both story and mechanics on top of that, it becomes harder to keep making better games of the same series (its especially difficult if the first game in the series was very good).
Some series IMO can get better over time while still having solid first releases, but those are few and far between, and usually are not long series. The Witcher series IMO is one of the few I can say where I genuinely think the games each got better off of each other in every respect. Visuals, story, gameplay, music, UI, you name it and I'll say it got better with each title release. And since The Witcher 3 ends Geralt's story (who knows what'll happen now with the franchise since CD Projeckt Red is stepping away from it for now to work on their Cyberpunk title), I'd call it a successful series from old to new.