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We are studying infinite sums right now in my calc class. They seem like a nice change of pace from the standard integration problems we had been doing. I can't yet see why your alternating and decreasing monotonic sequence converges when summed, but give me a week more of class and i will probably be able to.
Let's see:
1 - 1/2 = 1/2 = .5
1 - 1/2 + 1/3 = 5/6 = 0.83333
1 - 1/2 + 1/3 - 1/4 = 7/12 = 0.58333
1 - 1/2 + 1/3 - 1/4 + 1/5 = 47/60 = 0.78333
1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 = 37/60 = 0.61666
I don't have a general explanation as to why it converges, but based on its initial behavior it seems to be approaching a specific number.
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