And so I shall return to my favorite topic... mathematics
Consider the series: S = 1-2+3-4+5-6...
Method 1 for solving:
S = (1-2)+(3-4)+(5-6)+... = (-1)+(-1)+(-1)+... = -Infinity
Method 2 for solving:
S = 1-(2-3)-(4-5)-(6-7)+... = 1+1+1+... = Infinity
Hence, -Infinity = Infinity. Obviously wrong, unless you believe that the number line is not indeed a line, but a loop with an extremely large radius, hence appearing to be a line to us mortals.
What's the fallacy in the above argument?
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5 comments:
Dude,
You can only approximate the sum of an infinite series if the terms keep diminishing. After you actually sum up a large number of terms, a few additional terms make a very small difference to the sum, so at that point we make an approximation of sum to infinity.
In your case however, the terms keep increasing. So there is no such thing as approximate sum to infinity.
If you sum to a very large odd number N terms, your sum will be large positive, just add one more term your sum is very large negative. In the first case, you are considering an even number of terms and the second case an odd number of terms, I am not surprised by the results.
(The sum to infinity will be either positive or negative depending on whether infinity is odd or even :P)
In fact the sum to N terms of your series is given by CEIL(N/2)*POW(-1,N+1) and it varies like this
If you post math puzzles, only you and I will be talking about them, just like the puzzle thread we had in the group a little while ago :p
Guys, can you post some more puzzles? I don't always get the answers, but I like thinking about the problems.
Thanks.
Would be glad to oblige Anurag, I'm a puzzle enthusiast myself. What kind of puzzles though?
You might want to look at one of my earlier posts
I just tripped upon the formal solution to the problem at Wikipedia
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