Defining convergent and divergent infinite series
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Worked example: sequence convergence/divergence
So we've explicitly defined four different sequences here. And what I want you to think about is whether these sequences converge or diverge. And remember, converge just means, as n gets larger and larger and larger, that the value of our sequence is approaching some value. And diverge means that it's not approaching some value. So let's look at this. And I encourage you to pause this video and try this on your own before I'm about to explain it. So let's look at this first sequence right over here. So the numerator n plus 8 times n plus 1, the denominator n times n minus 10. So one way to think about what's happening as n gets larger and larger is look at the degree of the numerator and the degree of the denominator. And we care about the degree because we want to see, look, is the numerator growing faster than the denominator? In which case this thing is going to go to infinity and this thing's going to diverge. Or is maybe the denominator growing faster, in which case this might converge to 0? Or maybe they're growing at the same level, and maybe it'll converge to a different number. So let's multiply out the numerator and the denominator and figure that out. So n times n is n squared. n times 1 is 1n, plus 8n is 9n. And then 8 times 1 is 8. So the numerator is n squared plus 9n plus 8. The denominator is n squared minus 10n. And one way to think about it is n gets really, really, really, really, really large, what dominates in the numerator-- this term is going to represent most of the value. And this term is going to represent most of the value, as well. These other terms aren't going to grow. Obviously, this 8 doesn't grow at all. But the n terms aren't going to grow anywhere near as fast as the n squared terms, especially for large n's. So for very, very large n's, this is really going to be approaching n squared over n squared, or 1. So it's reasonable to say that this converges. So this one converges. And once again, I'm not vigorously proving it here. Or I should say I'm not rigorously proving it over here. But the giveaway is that we have the same degree in the numerator and the denominator. So now let's look at this one right over here. So here in the numerator I have e to the n power. And here I have e times n. So this grows much faster. I mean, this is e to the n power. Imagine if when you have this as 100, e to the 100th power is a ginormous number. e times 100-- that's just 100e. Grows much faster than this right over here. So this thing is just going to balloon. This is going to go to infinity. So we could say this diverges. Now let's look at this one right over here. Well, we have a higher degree term. We have a higher degree in the numerator than we have in the denominator. n squared, obviously, is going to grow much faster than n. So for the same reason as the b sub n sequence, this thing is going to diverge. The numerator is going to grow much faster than the denominator. Or another way to think about it, the limit as n approaches infinity is going to be infinity. This thing's going to go to infinity. Now let's think about this right over here. So as we increase n-- so we could even think about what the sequence looks like. When n is 0, negative 1 to the 0 is 1. When n is 1, it's going to be negative 1. When n is 2, it's going to be 1. And so this thing is just going to keep oscillating between negative 1 and 1. So it's not unbounded. It's not going to go to infinity or negative infinity or something like that. But it just oscillates between these two values. So it doesn't converge to one particular value. So even though this one isn't unbounded-- it doesn't go to infinity-- this one still diverges. It doesn't go to one value. So let me write that down. This one diverges.
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