Quote:
Originally posted by klaucke
No, that's not the case at all. Euler proved hundreds of years ago what the sum of the above geometric series is. I'm sure you can google it, the proof is quite simple, but I'd rather not write it out in Mathematica.
I just searched myself, you'll note Wikipedia has a quite involved article on it w/ numerous proofs.
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The second and third pictures show that an infinite series is defined as the limit of its sequence of partial sums as n tends to infinity. The first picture shows how the limit of a sequence is defined. The fact that epsilon is greater than 0 shows that the sequence never actually equals the limit S - it can just be made arbitrarily close by choosing a large enough N. So again, we are talking about a limit. The sum of the infinite series in your calculation approaches 1 but is not actually equal to 1. And 0.99999... with an infinite number of 9’s is still not equal to 1. Only 1 is equal to 1.
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