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Geometric Series - Concept
In Algebra, sometimes we have functions that vary in more than one element. When this happens, we say that the functions have joint variation or combined variation. Joint variation is direct variation to more than one variable (for example, d = (r)(t)). With combined variation, we have both direct variation and indirect variation.
A geometric series is what happens when we sum a geometric sequence, okay? A sequence is a series of numbers, the sum is always all added up together. And to find the sum of a geometric series we have a number of different equations at our disposal, okay?
So what we have is for a finite series, okay, that is a series with a set number of terms, we have these 2 equations at the top of the board. Oops, and I have misread them, miswritten them. This should be a a sub 1. Sorry about that. So, what we have is the first term times 1-r to the n over 1-r. This is the exact same thing as a sub 1 r to the n minus 1 over r to the 1, okay? These are opposite statements if you switch one of these, you switch the other. Negative ones cancel. So either one of these is perfectly fine, okay? Your book may have 1, just go with whatever your book has or your teacher tells you. Okay.
In general I will use this equation, okay? The first one. And the reason I do that is because this is the formula for a finite series. We also have another formula for a infinite series and basically that's one that never ends, okay? And the reason I chose this is because the denominator is going to be the same for both of these and not having to remember when to switch your denominator makes my life a little but easier, okay? So I'm going to use these 2, if you want to use these 2, that's perfectly fine as well. But basically what we have, so we have these 2 for finite and one for infinite.
One way you can tell the difference is for the finite one, you're summing a sub n okay, you are summing the first n terms, whatever that maybe. For the infinite series, we don't have an n. So that's telling us we don't have a specified term number which means we're summing everything, okay? There is one restriction though that we have to have when we are summing a infinite series, and that is that our absolute value of our rate has to be less than 1, okay? And what that means is that our terms have to be getting smaller, okay?
And I'm talking about positive negative because they can switch back and forth. But basically, the numeric part of our numbers have to be getting smaller. And how that actually works is I've written out this sequence right here. 8, 4, 2, 1, one half, one fourth, and basically what we're doing is dividing by 2 every time or multiplying by one half because we always have to multiply when finding our geometric sequences and series. And what happens, if we added up all these terms together, eventually the terms down here are so small they're not going to do anything. So our next term will be one sixteenth, one thirty second, one sixty fourth so on and so forth. Eventually those numbers when we're dealing with whole numbers aren't going to make a difference, okay. We add one one thousandths to a number we already have, it's not going to make a difference. So that's how this infinite series equation works. Okay? You're just counting on these numbers to eventually be so small they're not going to affect our sum. Okay?
So 2 really 3 different equations for summing a geometric sequence, a geometric series. Pick 2 or try to pick one of these 2 and then you have to remember this one as well. We have our finite and our infinite.