Calculus bc khan academy1/9/2024 One, it just feels like something that would be interesting to explore. We would call a series, of one plus 1/2 plus 1/3 plus 1/4, and you just keep adding For many hundreds of years, mathematicians have been fascinated by the infinite sum, which This difference between the p-series function being real-valued and the zeta function being complex-valued is critical, because it means that we can only use real analytic (i.e., calculus with real numbers) techniques with p-series functions but we can use complex analytic (i.e., calculus with complex numbers) techniques with the Riemann zeta function, which waters all sorts of sprouts of research in the field of analytic number theory to this day (whereas large interest in p-series kind of died out when Leonhard Euler solved the Basel problem in the 18th century), mostly in the form of the Riemann hypothesis, widely considered to be the greatest unsolved problem in pure mathematics. The Riemann zeta function, on the other hand, is defined as Then you would be all set.Yes, their technical definitions make them significantly different from each other, although it may not seem so at a glance. Terms here and you could prove this one diverges, Than or equal to zero and you want to prove it diverges, well, maybe you could tryĬorresponding terms are less than the corresponding Here is going to diverge, and if you have a, onceĪgain you know that all the b sub ns are greater So once again, if you wanted to prove that this thing right over Tells us if our smaller series diverges, if this one diverges, then the larger one must also diverge. Unbounded, each of these corresponding termsĪre larger, so this one must also be unbounded. So this would kind of be unbounded towards infinity. Oscillating between values the only way you could do that is if you had negative terms here, It's not going to diverge because it oscillates between two values. It's not going to go to negative infinity. This one right over here is the smaller, I guessĮach of its corresponding terms are smaller, what if you could prove this one diverges? Well if this oneĭiverges, it's going to go unbounded to infinity. Guess I could kind of put them in quotes, That the magenta series, the smaller one, and I Now what if you went the other way around? What if you could prove Series, each of the terms are non-negative. Of course it would only apply to the case where your original Prove that one converges, then you're good with this one. Large as the corresponding terms here, and if you can That whose corresponding terms are at least as Test tells us, well, just find another series Of have a gut feeling it converges, the comparison Gee, I wish I could prove that it converges, I kind Looking you have your a sub n and you're like So why is that useful? We'll see this in future videos. One that is in some ways I guess you could say To some finite value, then that tells us that the The ones here, if this one converges, if this oneĭoesn't go unbounded towards infinity, it sums Whose corresponding terms are at least as large as Us if, I guess in my brain the larger series, the one Proof here, but hopefully that gives you a little bit of intuition. Of bounded by this one, must also converge. It, or I guess when we think about it is kind The one that's larger, if this one converges, well then the one that is smaller than Than the corresponding terms here, but they're greater than zero, that if this series converges, Tells us that because all the corresponding terms of this series are less So n equals one, two, three, all the way on, and on, and on. Once again, this is true for all the ns that we care about. To the corresponding term in the second series. Terms in the first series are less than or equal Now, let's say we also know that each of the corresponding Infinity, because you don't have negative values here. Oscillate, because you're not going to have negative values here. That these are either going to diverge to positive infinity or they're going to converge So a sub n and b sub nĪre greater than or equal to zero, which tells us Is that all the terms in these series are non-negative. That's the series b sub nįrom n equals one to infinity, and we know some thingsĪbout these series. We're speaking in generalities here, and let's have another one. It's an infinite series from n equals one to infinity of a sub n. We are trying to decide whether a series isĬonverging or diverging. Get a basic understanding of the comparison test when
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