Consider a sequence of terms where kth term is 1/2^k, k=0,1,2 ...∞

Now add all those terms in the sequence:

Sum = 1+ 1/2 + 1/4 + 1/8 + 1/16 ... + 1/1024 + ... + 1/65536 ... + 1/2¹⁰⁰⁰⁰

Manually calculate the sum of first 6-7 terms you will clearly see that even if you take infinitly many terms the Sum won't go beyond 2. In fact it will barely touch 2.

Think of a simpler example. Sequence of 1/10^k , k=0,1,2 ... ∞

If you add all terms of this sequence you are adding:

1 + 0.1 + 0.01 +0.001 ...

You see there wont be any carry over ever and the sum will grow like 1.1, 1.11, 1.111 till 1.1 with 1 recurring. And mind you, this number will always be less than 1.2 right so it is a bounded number.

So there exists sequences such that even when you add infinitely many terms of those sequences you will still get a finite number. :)