Zacay100 began this thread with the following post
I was wondering, I just read the book,]why could gollum see frodo when frodo was wearing the ring? He was invisible], and gollum could fight him!??! And also took a bite of his hand.
If he could smell him, still he knew exactly where to bite.
Good question Zacay100 and welcome to our forum.
My take on it is that Gollum saw where Frodo stood when he placed the ring on his finger. Gollum could still hear him and possibly even see Frodo's shadow which may have been cast by the fire. So he just ran to this area and grabbed. By the "Grace of Eru" he happened to grab the wrist of Frodo's ring-hand, and from there on, the ending was academic.
This was just another of those critical coincidences that leads me to believe that Eru was working in the background (The Music of the Ainur) towards the destruction of Sauron and the ascendancy of Man.
Some of those other critical coincidences were:
Gildor's Elves showing up to scare off the first Ringwraith and befriend the hobbits at Woody End.
Tom Bombadil showing up to rescue the hobbits from Old Man Willow.
Tom Bombadil showing up to rescue the hobbits from from the Barrow-Wight and providing Merry his Blade of Westernesse, so he could later distract the Nazgul Lord's attention away from Eowyn, so she could fulfill the original Glorfindel's prophesy).
Glorfindel (not Arwen) and Asfaloth meeting them on the road to the Ford of Bruinen.
Treebeard meeting Merry and Pippin in Fangorn Forest.
Faramir capturing Frodo and Sam in Ithilien.
Gorbag, Shagrat, and their troops fighting over Frodo's mithril coat.
Guri-Ghan-Guri's leading Theoden through the Stonewain Valley.
The wind shifting allowing Aragorn's fleet to sail upstream to the Pelennor Fields.
The meeting and argument between the two Orc bands at Isemouthe.
The Great Gollum's Banana Peel Ballet.
Gwaihir, Landroval, and their vassels showing up just when Gandalf needed them to rescue Frodo and Sam from Mt. Orodruin.
There are probably a few more, but you can get the idea from these.