I also think it converges................i.e.............that it has a limit.........the halving series.............probably adds up to 1.9999999........................b/c if its limit is 2................................then.......like in percentages..............less than 100%.........but almost 100%..............is like.................99.999999% etc..................








“Nonsense, you gullible old toad!” you are perhaps shouting to your screen. “Why should throwing out the numbers with 9’s make such a difference? We’ve still got all the other numbers!”
Again, I say: not so fast. You’re making a classic mistake. When you think of “numbers,” you’re only picturing little numbers.
“No I’m not!” you may say. “I’m thinking of big numbers. Huge numbers. Like 9 million, or 47 billion, or 228 trillion.”
Exactly my point: small numbers.
You see, the longer a number gets, the harder it is to avoid a 9. Every time you add a digit, you add a new opportunity for a 9. That might not feel like a big danger—after all, those 9’s will pop up only 10% of the time. But look what happens:

It’s as if, every time you type a digit, there’s a 10% chance that a giant numeral “9” falls from the sky and crushes you. For short numbers, with just 2 or 3 digits, you’re not very worried. The chances are in your favor.
But now you begin to type a 100-digit number. How do you like your chances? Sure, that giant “9” probably won’t fall on this digit… nor on this digit… but how long do you think your luck will last? Eventually you’ll get unlucky, and that “9” will come plummeting from the sky.

In the long run, most numbers have 9’s in them. Virtually all of them, in fact. So if you throw out these numbers from the harmonic series, it’s no surprise that it now converges. You’ve thrown out almost the entire series!
Want to know the weirdest part? The same logic applies to a longer sequence of digits. Say, 999.
At first, most numbers won’t have this sequence. But picture a billion-digit number. Just writing this number out—in size 8 font, double-siding the printing to save trees—takes a stack of paper as tall as a house.
Surely somewhere in there the number is bound to have the digits “999” in that order, right?
The same is true of every equally big number. And MOST numbers are this big! After all, there are only finitely many smaller numbers, and infinitely many bigger numbers. So, again, we’re throwing out almost the entire series.
Thus, the harmonic series also converges if you throw out all the numbers that include the digits “999.”
Perhaps you can see where this is going. (If so, you may be experiencing vertigo and/or nausea; this is normal.) The logic above works for any string of digits you can possibly think of. Even, say, a sequence of a million 9’s in a row.
So, in conclusion, the harmonic series diverges. It eventually outgrows any ceiling you’d put on it.
But simply throw out the numbers that happen to include a string of a million 9’s in a row… and suddenly the series converges. It plateaus. There’s some value that it will never surpass.
Somehow, by excluding only those numbers with a million 9’s in a row, you’ve changed the nature of the series.
In the words of Weinersmith: