Quote:
Originally posted by T_Samner
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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While what you posted is somewhat true in general, that is not the case here. There is an explicit formula for the sum of an infinite geometric series as I posted above. You do not need to take any limits w/ n,N, and epsilon in order to derive the explicit form. Like I said, check around, it's been known for hundred of years. Just because a sequence never reaches it's limit, does not mean the series of sums defined by it doesn't when taken ad infinitum.
Wikipedia has a proof for the geometric series here:
http://en.wikipedia.org/wiki/Geometric_series
And here is an article on .999... = 1: