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Unformatted text preview: MTH 122 — Calculus II Essex County College — Division of Mathematics and Physics 1 Lecture Notes #15 — Sakai Web Project Material 1 Don’t Be Misled! Last time we showed that S = 10 + 100 + 1000 + ··· was negative, and we all agreed 2 that was clearly wrong, although our reasoning was quite sound. Basic to our understanding was that we are trying to extend our understanding of the finite to the infinite, without having a real grasp of how difficult this actually is. So, it is clearly easy to be mislead when dealing with the infinite, especially if we think our finite ways can be easily extended to the infinite. They can’t! The last example is quickly moving towards the infinite, and this negative result we derived is clearly wrong. Likewise, you should also note that we showed that 1 = 0 . ¯ 9 by looking at the infinite sum, it was geometric. Anyway, don’t be misled into thinking that the sums of successively shrinking terms will always work out to be a finite number. In fact a very famoussuccessively shrinking terms will always work out to be a finite number....
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