eli5 how does the sum of infinite terms turns to be a finite value

Question

does this apply only for terms that are less than 1 eg 1 + 1/2 + 1/4…. or does this apply to all ap/gp. I remember studying this, but it’s been so long I remember only the gist.

In: 248

Answer 1

Let’s do a really simple one:

1/10+1/100+1/1,000+1/10,000…

In other words: .1 + .01 +.001 +.0001…

What do we get? Obviously 0.1111111… (repeating), which is converges to 1/9.

So in this simple case we can easily see that continually adding an extra 1 after the decimal point has an upper bound. No matter how many times we ad an extra 1 after the decimal, the sum quanity will never even reach 1.2

We can extend this concept intuitively to say generally that if the successive terms shrink sufficiently quickly, they are so many decimal places removed from the previous terms that they can never “catch up”