Infinite Series

rayroshi

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Mar 14, 2011
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I need help understanding why it is that some infinite series will diverge to infinity, while others will converge to a specific limited value/number, even though both series consist of successive terms that are positive, decreasing values; e.g., the harmonic series 1/1 + 1/2 + 1/3 + 1/4 + ... diverges to infinity, yet the series 1 + 1/2^2 + 1/3^2 + 1/4^2 + ... converges. Both contain terms that are positive and decreasing. I can see why the first series diverges to infinity, since an infinite number of values logically adds to an infinite amount; however, I cannot see why the second series shouldn't do the same for the very same reason. In both cases/series, the terms are heading toward zero.

It would be much appreciated if any attempted answer would be couched in a plain-language format, rather than in complex mathematical "formulese." It has always been my experience that, if someone can't put an explanation into clear, simple terms, they probably don't understand it that well, themselves. That observation obviously has its limitations, as I'm sure it would be rather hard to put such things as advanced quantum physics concepts into simple terms, but I don't think we're talking about such a lofty level, in the case of my low-level math skills, struggling with infinite series, here.
 
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stapel

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I need help understanding why it is that some infinite series will diverge to infinity, while others will converge to a specific limited value/number, even though both series consist of successive terms that are positive, decreasing values; e.g., the harmonic series 1/1 + 1/2 + 1/3 + 1/4 + ... diverges to infinity, yet the series 1 + 1/2^2 + 1/3^2 + 1/4^2 + ... converges. Both contain terms that are positive and decreasing. I can see why the first series diverges to infinity, since an infinite number of values logically adds to an infinite amount; however, I cannot see why the second series shouldn't do the same for the very same reason. In both cases/series, the terms are heading toward zero.
But one is heading toward zero "faster" or "more strongly". Some of them power toward zero strongly enough to topple over to the ground and settle onto some finite value. Others edge toward zero, but in a weak, ambling manner, so that they're able to escape the pull of the fixed finite numbers, always skimming along, just high enough above the ground to "add up to infinity", eventually. ;)
 

rayroshi

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I'm a 'little' late in responding, here (over six years), and I apologize for that. But I must say your answer was a good one that fit the level of technicality that I was looking for. Thank you.
 

Singleton

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6 years? And you are still interested in the question?

To convince yourself that an infinite sum of positive numbers can be finite, just take a 1x1 square, area is 1. Divide it into rectangles areas 1x(1/2) , take one of these and divide into squares of area (1/2)x(1/2). You can keep going and add the infinite number of little areas. But they still add up to 1.
 

lookagain

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... the harmonic series 1/1 + 1/2 + 1/3 + 1/4 + ... diverges to infinity, yet the series 1 + 1/2^2 + 1/3^2 + 1/4^2 + ... converges. Both contain terms that are positive and decreasing. I can see why the first series diverges to infinity, since an infinite number of values logically adds to an infinite amount; however, ...
No, it does not follow. For example, a geometric series of the type
such as 1/10 + 1/10^2 + 1/10^3 + ... = 0.111... = 1/9 is convergent.
 

rayroshi

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6 years? And you are still interested in the question?

To convince yourself that an infinite sum of positive numbers can be finite, just take a 1x1 square, area is 1. Divide it into rectangles areas 1x(1/2) , take one of these and divide into squares of area (1/2)x(1/2). You can keep going and add the infinite number of little areas. But they still add up to 1.
Ha! That was a great example. Thanks for sending it.
 
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