28. The modern world is full of people who know just enough to be able to discus
ID: 2886656 • Letter: 2
Question
28. The modern world is full of people who know just enough to be able to discuss a topic, but not enough to deal with that information correctly. Here is an example involving infinite series. If you don't handle them with care, you can get some truly wacky results. Consider the following "proof that 0 1. -1-1+1-1+1-1+. 1+01+ There must be an error in this "proof' somewhere! But at first glance, it looks fine. Try your best to pinpoint the error's exact location and explain clearly what went wrong. asoning, even with Comment: The point of this problem is to show you how careful you must be when re something as basic and objective as math. If someone makes a good argument that leads to a conclusion you think is probably false, examine that argument carefully. As this problem shows, it can be easy to mistake and yet hard to pinpoint it. Better yet, be willing to question your own reasoning. Careul thinking will almost always uncover the truth!Explanation / Answer
0 = 0 + 0 + 0 + ........----------fine (say total n zeros)
= 1 - 1 + 1 - 1 + 1 - 1 + ...... ------- fine, one 0 is repleced by (1-1) pair , so total number terms or 1s is now 2n
= 1 + (-1+1) + (-1+1) + .... ------now... at the end of the previous line there was (1 - 1) pair, but in this line there
is a seprate '-1' at the end as we have taken away the +1 from it's pair. total number of 1's still should be 2n (even). count total 1s in this line-> 1 + 2 + 2 + 2 + 2.......and +1(the last '-1' without pair)
= 1+ 0 + 0 + 0 + ...... -1 ---- mistake is here. we forgot that '-1' at the end of the line and made all the terms 0
after 1 at first position. by mistake, we took total number of 1s odd rather than even.
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