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.

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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