 Here's a better argument than me smacking the tally upside the head with a baseball bat there I did get it on the record just for you pizza To eat a pizza you can start out by eating half of the pizza But you can take that pizza and divide it in half again that half of a piece In fact, you can keep dividing it to smaller and smaller and smaller slices Which would suggest that there's an infinite number of slices But is it possible for a group of people to finish off an entire pizza? Yes So it seems that there are some series that go forever That reach a total We say they converge and there are some series Brett that go forever That don't reach a total here's an example of a series that would never Reach a total two four eight sixteen thirty two sixty four are my numbers just getting bigger and bigger and bigger and bigger I think when you add up bigger and bigger and bigger and bigger and bigger and bigger stuff I think that eventually reaches or approaches infinity But what Xeno suggested in his little argument that I started this class out with is if you're adding up smaller and smaller And smaller and smaller stuff in some cases You'll approach and answer So David buys a pizza he eats half of the pizza then half of the remaining piece then half of half of the remaining piece etc Continuing in this way it looks like he will never finish the pizza because there will always be half of a piece left Explain why in practice this could not happen. Well, you can eat a pizza Says represent this situation with a geometric series whose first term is a half Normally, I would love to walk through all this but because I'm hoping to go over the trig test with you a bit later on in class I'm gonna say turn to page 183 page 183 and what I'm going to do is I'm going to write two series the first one Brandon give me number between one and ten five comma 15 comma. Oh, but instead of commas. I'm sorry. Let's put plus signs plus 15 plus What am I multiplying each term by? 45 plus 135 dot dot dot Here r equals three What do you notice about your terms each successive term is getting bigger and bigger and bigger and bigger Compare that with series number two. I'll start with oh heck five comma oh Not comma plus doing that. Sorry one fifth plus 125th plus one one hundred and twenty fifth dot dot dot What's our in this case? How can I find our always? Any term divided by the one in front of it? Oh, you know what I made a mistake in my series, didn't I? Cross out the five make it a one Is right, you know what r is in this case folks? If I've done the math correctly R is one fifth Amy. What do you notice about the terms here? Are they getting bigger or are they getting smaller? smaller We say this first series here diverges We say this second series here Converges and you can tell just by looking at our if our is a fraction Your series will converge to a given value eventually if our is a number bigger than one not a fraction Your series will diverge we're gonna write that a bit more formally in Part B Okay, it says this your sequence will converge is convergent and Approaches the value it says zero if our Blank comma blank. You know what I like this statement here R will converge if your Ratio is between negative one and positive one if it's a fraction, but it could be a negative fraction, Dina The easier way to write that is to say if the absolute value of r is less than one because that includes a negative or the positive R will get bigger r to the end will get bigger which means your series will diverge if r is bigger than positive one or less than negative one or Absolute value r is greater than one it seems confusing for a little bit patience young grasshoppers turn the page provided That r is between negative one and positive one. How can you figure out our any term divided by the one in front of it? You can actually add up an infinite number of terms using This year what does letter s stand for what did it stand for last unit? The sum except last lesson Elizabeth it was s with a little n right there saying the number of terms You notice there's no subscript there. It's saying all of them if you keep going to infinity It's the first term Divided by one minus r. There is a lovely proof right there. I just don't have time to walk through it I do have time to show you how to use this so here's The geometric series we've learned so far if it's a finite series if I tell you how many terms or nick I tell you the last term use one of these two You know what I like that l a bit better than the one on the previous lesson which looked like a one at least that stands out if R is a fraction between negative one and positive one or the absolute value of r is that you can find the infinite sum The same way as I was able to smack the tally What was r in my example a half with the tally because I kept saying half of a half of a half of half of half of half If r is bigger than one or less than negative one or the absolute value of r is bigger than one It won't converge. You could only find a finite sum Let's use this page 185 Example one it says find the common ratio for each of the following geometric series. Let's do that first What's r here? One third. Oh, hopefully you were clever enough to go one third divided by one not one ninth divided by one third Even though you get the same answer What's r here negative five yes? What's r here negative a half Which of these will converge? Which of these can you actually take an infinite sum of which of these have common ratios? That are fractions less than one Okay, so this one here diverges This one here is going to be The infinite sum. I'll even put a little infinity as my subscript, which is an eight on its side You don't have to you can leave it totally blank, but it's going to be a all over one minus r What's a? What's a one? What's one minus r in your head, please? What's one minus one-third? Two-thirds you can do some of them in your head in fact I could now say how do I divide by a fraction flip it and multiply I could put this one on a non-calc section and feel comfortable with it But the answer on your calculator or how do I divide by fractions of the multiply is three over two or 1.5 is it not Get your calculator out all of you If you don't have a calculator sheepishly come get one no takers. Okay, good calculator out I Want you to notice how fast this does approach 1.5 so go one plus One-third and hit enter and then add one ninth and hit enter and Then add what would come after one nine? Can you do the math? 127 and hit enter plus what would come after one over 27? one over three to the fourth 81 Plus one over three to the fifth. I'm just continuing the pattern Plus I mean we're at 1.5 basically right now Plus and we're only on our what sixth term fifth term it converges pretty fast Plus one over three to the sixth would be the next one Plus one over three to the seventh would be the next term. Here's my I've only added now I believe eight terms and I'm at one point four nine nine They converge pretty fast think about that The first eight terms gave you one point four nine nine the remaining infinity terms Only add up to point zero zero zero two two eight six two three six eight five six dot dot dot The nerd within me finds that kind of cool infinity behaves weird Be won't converge it won't have an infinite sum See will let's find it. So the infinite sum is equal to a all over one minus r You know what on the previous page in keeping with my tradition since this is on your formula sheet I should have put a big highlighter around that. Yes for God and It's weird that my brain twink me right then right now. Don't ask me how my brain works. It's frightening sometimes If you continue this pattern forever and kept adding what would you get? Let's see. It's gonna be two all over one minus Minus a half that's the same as one plus a half. You know what one plus a half is three over two of one point five The answer that the sum of infinity There are other types of series that converge But there's a whole sub branch of math Which does nothing but try and prove ahead of time mathematically which series converge and which series diverge Who's in calculus? Really important in limits So in probably second or third year count you'll start taking the limit of a series Not nice your easy ones like this, but of a series where oh the denominator is actually a function an equation. Oh cool or not example two The first term of a geometric series is to oh a equals two and The sum to infinity is four. I Guess that means that s equals four What's what do they want me to find here are? How do I know which s formula to use the one from last day or the one today? Well the key is they said the sum to infinity Which means I'm going to use the infinite sum equation, but I need a bit more room Let's plug stuff in what's s the sum to infinity for Equals what's a to all over one minus r can anyone tell me how to solve this please please please oh Yeah Good old cross multiply we'll get To equals for bracket one minus r To equals I could divide by four. I'm gonna go to brackets here Four minus four are the reason is Brandon. I've done this style of equation so much since grade eight I think now I can do this in my mind minus four from both sides and then divide by negative four. Yes R is gonna end up being to take away for Divided by negative four. Oh and lowest terms. I guess R is a half Now three a and b is actually a neat application. You can use this to write fractions that go on forever Because this is really point zero seven plus point zero zero seven plus point zero zero zero seven plus point zero zero zero seven where R is one-tenth and Then you can solve for s and you'll get an answer as a fraction and now you go. Oh, that's what that was as a repeating decimal I Sadly don't have time because I want to go through your trig test so very quickly homework 1a b C m three four three and four are mini curveballs seven a Part one two and three, but don't worry about b and nine