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KnifeMissile · 10-25-2004, 01:09 PM

Actually, this is a precursor to a post that I really want to make and, actually, I've wanted to make this post for some time now but haven't gotten around to it and now it looks like the best days of this forum are gone. I don't recognize all the interesting characters who could appreciate what I'd like to explain. Oh well, maybe they're still lurking. Here goes...

...the precursor, that is. Are we all aware of what an infinite sum is?

The harmonic series is a series of the form: 1 + 1/2 +1/3 + ... + 1/n. Surprisingly, even though the terms limit to zero, this series doesn't converge. It will sum indefinitely high. However, if you alternate the terms, like so: 1 - 1/2 + 1/3 - ... + (-1)^n/-n (order of operations apply), then this does converge! In fact, it can be shown that any series, where the term sign alternates and the absolute values of the terms monotonically decrease, will converge to a finite value. Can everyone see why this is true? Can a stronger statement be made?

The claim I'd like to make will probably leave most physicists incredulous. I've tried to explain it to some before but have only been met with skepticism. Hopefully, many math enthusiasts will undestand me. For now, I just want to know what the board is currently like...

10-25-2004, 01:09 PM	#1 (permalink)
KnifeMissile Location: Waterloo, Ontario	Queer properties of infinite sums... Actually, this is a precursor to a post that I really want to make and, actually, I've wanted to make this post for some time now but haven't gotten around to it and now it looks like the best days of this forum are gone. I don't recognize all the interesting characters who could appreciate what I'd like to explain. Oh well, maybe they're still lurking. Here goes... ...the precursor, that is. Are we all aware of what an infinite sum is? The harmonic series is a series of the form: 1 + 1/2 +1/3 + ... + 1/n. Surprisingly, even though the terms limit to zero, this series doesn't converge. It will sum indefinitely high. However, if you alternate the terms, like so: 1 - 1/2 + 1/3 - ... + (-1)^n/-n (order of operations apply), then this does converge! In fact, it can be shown that any series, where the term sign alternates and the absolute values of the terms monotonically decrease, will converge to a finite value. Can everyone see why this is true? Can a stronger statement be made? The claim I'd like to make will probably leave most physicists incredulous. I've tried to explain it to some before but have only been met with skepticism. Hopefully, many math enthusiasts will undestand me. For now, I just want to know what the board is currently like...