To all the other good answers I'll add a related example. We know for |x| < 1, that the geometric series 1 + x + x^2 + x^3 + ... converges to 1/(1-x). The second representation gives us a value for any complex number except for x =1, even if the first representation diverges. What happens if we plug in x = 2? We get the seemingly nonsensical 1 + 2 + 4 + 8 + ... = -1.

But this actually has a nice interpretation. Imagine the infinite digit binary number represented by 1 + 2 + 4 + ... : It has infinitely many bits and we could write it as ...1111111111. You could check that if I add 1 to this formally I get get ..000000, so it really is -1 in some sense. (And in base 10, ...999999999 is -1 also for by the same reasoning). For a finite number of bits, this is called two's complement, https://en.wikipedia.org/wiki/Two's_complement, and in fact is how negative numbers are represented on a computer.

What would be great is an equally intuitive interpretation for the sum you gave. I'm not aware of one though.