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These are the user uploaded subtitles that are being translated: 1 00:00:00,180 --> 00:00:04,010 In this video we're going to be talking about how to find the sum of a geometric series. 2 00:00:04,050 --> 00:00:08,850 And if we want to find the sum of a geometric series we first have to be able to determine that the 3 00:00:08,850 --> 00:00:14,430 geometric series converges and in order to say that the geometric series converges of course we have 4 00:00:14,430 --> 00:00:18,960 to be able to say that we are in fact dealing with a geometric series instead of some other kind of 5 00:00:18,960 --> 00:00:19,330 series. 6 00:00:19,320 --> 00:00:24,000 So working backwards that way we're going to kind of think about this problem in three steps we've been 7 00:00:24,000 --> 00:00:29,130 given this series and we haven't been told explicitly it's a geometric series we're going to find that 8 00:00:29,130 --> 00:00:29,740 out a second. 9 00:00:29,760 --> 00:00:34,920 But what we want to do is first show that this is in fact a geometric series which will do by getting 10 00:00:34,920 --> 00:00:40,210 the series into this form here then we'll say whether or not the geometric series converges. 11 00:00:40,290 --> 00:00:43,630 And then if it does converge we'll find some using this formula. 12 00:00:43,860 --> 00:00:47,520 So first of all let's talk about how to say whether or not this is a geometric series. 13 00:00:47,640 --> 00:00:51,340 If it is a geometric series we'll be able to get it into this form here. 14 00:00:51,390 --> 00:00:56,970 The infinite sum from end equals zero to infinity of a Times are to the end power and we'll come back 15 00:00:56,970 --> 00:00:57,960 to that in a second. 16 00:00:58,030 --> 00:01:04,350 The value we've been given is this infinite or repeating decimal zero point to one with the bar over 17 00:01:04,350 --> 00:01:07,260 the two one which means that the 2 1 repeats. 18 00:01:07,260 --> 00:01:14,870 So this is we can say the same thing as zero point two one two one two one two one repeating forever 19 00:01:14,880 --> 00:01:16,850 so that to 1 just repeats. 20 00:01:16,980 --> 00:01:20,450 So we're looking to do is turn this into a geometric series. 21 00:01:20,460 --> 00:01:25,960 And what we want to recognize is that we have each of these two one pairs. 22 00:01:26,130 --> 00:01:28,250 We can turn into a fraction. 23 00:01:28,260 --> 00:01:31,200 So if we just look at this point to one here. 24 00:01:31,380 --> 00:01:39,330 We know that that's the same as twenty one over 100 so we can call that 21 over 100. 25 00:01:39,330 --> 00:01:43,400 We know it's over 100 because this number ends in the hundredths place. 26 00:01:43,410 --> 00:01:47,100 The two is in the 10th place the one is in the hundredths place. 27 00:01:47,100 --> 00:01:53,450 So since this point to one here ends in the hundredths place it's the same as 21 over 100. 28 00:01:53,670 --> 00:01:56,670 What about the second 21 here. 29 00:01:56,940 --> 00:02:03,570 This 2 one while the two is in the thousands place the one is in the 10000 place. 30 00:02:03,570 --> 00:02:14,130 So what we want to do is add to this 21 over 100 we want to add 21 over ten thousand because 21 over 31 00:02:14,130 --> 00:02:17,130 10000 represents this to one here. 32 00:02:17,160 --> 00:02:23,870 And if we add these two fractions together we get Point 2 1 to 1 to deal with the next 21 here. 33 00:02:23,940 --> 00:02:30,930 This 21 the two is going to be in the 100000 place which means the one is going to be in the millionths 34 00:02:30,930 --> 00:02:31,640 place. 35 00:02:31,650 --> 00:02:34,950 So what we want to do is go ahead and add it to this. 36 00:02:34,950 --> 00:02:45,280 Some here we want to add 21 over 1 million and noticed that we're just adding two zeros to the denominator 37 00:02:45,310 --> 00:02:50,330 every time and that should make sense because we have a number here that repeats it's a two digit number. 38 00:02:50,350 --> 00:02:56,320 So once we identify this 21 over 100 We know that the next 21 since it's two digits it's just going 39 00:02:56,320 --> 00:02:59,590 to have two more zeros here in the denominator so we add two zeroes. 40 00:02:59,590 --> 00:03:02,010 We add two zeros and we could do that forever. 41 00:03:02,110 --> 00:03:07,750 For now we're just going to say plus dot dot dot because this goes on forever and we're just looking 42 00:03:07,750 --> 00:03:10,840 for a sum that represents this repeating decimal. 43 00:03:10,840 --> 00:03:17,120 So once we have it in this form what we can do is factor out a 21 over 100 from this sum. 44 00:03:17,140 --> 00:03:25,240 So if we factor out 21 over 100 we have to multiply 21 over 100 by one to get this 21 over 100 then 45 00:03:25,240 --> 00:03:26,780 we're going to add to that here. 46 00:03:26,830 --> 00:03:31,820 What do we have to multiply by 21 over 100 to get 21 over 10000. 47 00:03:31,990 --> 00:03:36,640 Well we have to multiply the numerator by 1 21 times 1 gives us 21. 48 00:03:36,700 --> 00:03:38,470 So the numerator is going to be 1. 49 00:03:38,470 --> 00:03:42,010 What do we have to multiply by 100 to get 10000. 50 00:03:42,040 --> 00:03:46,830 Well that's 100 100 times 100 gives us 10000. 51 00:03:46,840 --> 00:03:51,340 And remember you can just add the zeros here and 10000 we have 1 2 3 4 zeros. 52 00:03:51,340 --> 00:03:57,970 That means we have to have 1 2 3 4 zeros between these two to get four zeros we multiply them together. 53 00:03:58,090 --> 00:04:03,220 So we get plus 1 over 100 and then we're going to get plus 1 over 100. 54 00:04:03,220 --> 00:04:05,450 And again we just add two more zeros. 55 00:04:05,500 --> 00:04:11,050 So one hundred times 10000 gives us 1 million and we'll go ahead and say plus dot dot dot. 56 00:04:11,170 --> 00:04:14,890 And now we factored 21 over 100 out of the sum. 57 00:04:14,890 --> 00:04:20,460 Now if we just look at the values here inside the parentheses if we keep this 21 over 100 out in front 58 00:04:20,590 --> 00:04:23,300 if we look at the values inside these parentheses here. 59 00:04:23,410 --> 00:04:25,390 Let's start with the one over 100. 60 00:04:25,420 --> 00:04:30,880 Well that's the same thing as saying one over 100 raised to the first power one over 100 raised the 61 00:04:30,880 --> 00:04:32,840 first power is just 1 over 100. 62 00:04:32,840 --> 00:04:40,180 So thinking one over 10000 is the same thing as quantity 1 over 100 squared. 63 00:04:40,180 --> 00:04:43,570 Remember anything raised to the power of 0 is just one. 64 00:04:43,570 --> 00:04:49,750 That means that we can represent one here with one over 100 raised to the zero power. 65 00:04:49,750 --> 00:04:54,060 So what you can see here then is that we're starting to get this pattern here. 66 00:04:54,070 --> 00:04:57,750 One over 100 raised to the zero power the first power the second power. 67 00:04:57,910 --> 00:05:01,960 Which means if we went on forever the next turn would be 1 over 100 raised to the third power Fourth 68 00:05:01,960 --> 00:05:02,870 Power fifth power. 69 00:05:02,920 --> 00:05:05,950 And so what's starting to emerge is a geometric series. 70 00:05:05,950 --> 00:05:12,730 Because if we compare this series to this infinite sum equals zero to infinity of a R to the N. 71 00:05:12,810 --> 00:05:20,080 But we can recognize here is that a is a coefficient that's multiplied by this art of the N series. 72 00:05:20,100 --> 00:05:27,260 Well in our case is just 21 over 100 it's a value that's multiplied out in front by this series here. 73 00:05:27,370 --> 00:05:32,680 Notice here than in this series and starts at zero which means we're looking for art of the zero plus 74 00:05:32,900 --> 00:05:34,670 the first Plus are the second. 75 00:05:34,690 --> 00:05:40,560 Well that looks awfully similar to this one over 100 0 plus 1 over 100 for the first plus 1 over 100 76 00:05:40,560 --> 00:05:42,330 to the second et cetera. 77 00:05:42,340 --> 00:05:49,610 So then what we identify is that this one over 100 value is the same as are here. 78 00:05:49,810 --> 00:05:55,890 And that because START starts zero we see and starting down here 0 1 2 and counting up. 79 00:05:55,900 --> 00:06:01,840 So essentially by getting the series into this forum we've identified a value for A and a value for 80 00:06:01,840 --> 00:06:02,510 our. 81 00:06:02,530 --> 00:06:08,380 Once you get the series to this point here you don't have to go to the next step you can identify and 82 00:06:08,440 --> 00:06:15,300 are right at this point a is always going to be this value out in front of the parentheses. 83 00:06:15,610 --> 00:06:18,470 Is always going to be this second term here. 84 00:06:18,550 --> 00:06:21,310 The turn immediately following the 1. 85 00:06:21,370 --> 00:06:26,560 So as long as the series Inside these parentheses here starts with 1 which is why we factored out the 86 00:06:26,560 --> 00:06:31,720 21 over 100 because we wanted this to start with one as long as we've gotten this to start with one 87 00:06:31,900 --> 00:06:33,830 then that means that this is going to be a. 88 00:06:34,030 --> 00:06:39,460 And this is going to be our Assuming of course that you get are the first R-squared are the third are 89 00:06:39,490 --> 00:06:42,510 the fourth etc. which in this case we do. 90 00:06:42,610 --> 00:06:45,510 So we've identified a and we've identified our. 91 00:06:45,670 --> 00:06:50,710 And we've shown that this series has the same form as the series right here which means that this is 92 00:06:50,710 --> 00:06:54,620 in fact a geometric series because it has the same form as this series. 93 00:06:54,760 --> 00:06:58,720 And we can identify n are we know that this is a geometric series. 94 00:06:58,720 --> 00:07:04,450 So a sense of the geometric series we can use this geometric series Test to say whether or not the series 95 00:07:04,450 --> 00:07:10,030 converges according to the geometric series test a geometric series will converge with the absolute 96 00:07:10,030 --> 00:07:12,030 value of r is less than 1. 97 00:07:12,040 --> 00:07:15,280 Well we know in this case that our is 1 over 100. 98 00:07:15,280 --> 00:07:16,920 We've identified that already. 99 00:07:17,050 --> 00:07:21,890 So we can say the absolute value of 1 over 100 less than 1. 100 00:07:21,910 --> 00:07:26,530 And in fact that is true there there's no negative sign here so the absolute value of 1 over 100 is 101 00:07:26,530 --> 00:07:29,960 just 1 over 100 and that is less than 1. 102 00:07:30,010 --> 00:07:37,270 So the series will in fact converge if this value one over 100 was greater than or equal to 1. 103 00:07:37,330 --> 00:07:42,230 We would know that the geometric series was divergent but because it's less than one we can say that 104 00:07:42,230 --> 00:07:46,090 this particular geometric series converges or that it's convergent. 105 00:07:46,160 --> 00:07:51,410 So because it converges we can find it some if it was divergent we wouldn't be able to find the sum 106 00:07:51,440 --> 00:07:53,900 but because it's convergent we can find some. 107 00:07:53,900 --> 00:07:57,150 So what's the sum of this geometric series. 108 00:07:57,150 --> 00:08:01,990 Well the sum is always given by this formula a divided by 1 minus r. 109 00:08:02,090 --> 00:08:05,900 And when we say the same here what we're talking about is this. 110 00:08:05,930 --> 00:08:06,960 Some over here. 111 00:08:06,980 --> 00:08:12,690 So this whole sum right here the sum from N equals zero to infinity of a part of the N. 112 00:08:12,740 --> 00:08:18,140 This whole series if we want to find the sum of this series the sum is going to be a over 1 minus or 113 00:08:18,140 --> 00:08:24,740 so sometimes you'll see the formula where you'll see this infinite sum set equal to a over 1 minus R 114 00:08:24,980 --> 00:08:26,590 because they're equal to one another. 115 00:08:26,690 --> 00:08:32,810 So in order to find the sum we're just going to plug in a and r to this a over one minus our formula 116 00:08:32,840 --> 00:08:34,250 and then we're going to simplify. 117 00:08:34,400 --> 00:08:35,860 So we know that a. 118 00:08:35,870 --> 00:08:38,260 In our case is equal to 21 over 100. 119 00:08:38,330 --> 00:08:44,330 So we're going to put that in the numerator and then in the denominator we're going to get 1 minus r 120 00:08:44,450 --> 00:08:48,690 r we know is 1 over 100 so we're going to get 1 over 100. 121 00:08:48,770 --> 00:08:50,830 And now we want to go ahead and simplify this. 122 00:08:50,930 --> 00:08:53,230 And you're always going to simplify in the same way. 123 00:08:53,240 --> 00:08:58,850 So instead of one here in the denominator we're going to find a common denominator with this fractions 124 00:08:58,850 --> 00:09:03,390 instead of one we're going to get 100 over 100 that's the same thing as 1. 125 00:09:03,410 --> 00:09:09,550 That way we can combine these two fractions in the denominator and say 100 minus one is ninety nine. 126 00:09:09,650 --> 00:09:16,860 So this becomes 21 over 100 divided by 99 over 100. 127 00:09:16,970 --> 00:09:21,610 And I remember that when we have a fraction divided by a fraction we keep the fraction in the numerator. 128 00:09:21,620 --> 00:09:23,280 21 over 100. 129 00:09:23,360 --> 00:09:28,820 And instead of dividing by the fraction from the denominator we multiply by it's reciprocal. 130 00:09:28,820 --> 00:09:33,470 So instead of division we'll do multiplication and the reciprocal is whatever we get when we flip this 131 00:09:33,470 --> 00:09:34,460 fraction upside down. 132 00:09:34,460 --> 00:09:40,500 So instead of 99 over 100 we'll say 100 over ninety nine and we haven't changed anything. 133 00:09:40,550 --> 00:09:43,600 This multiplication is the same thing as this division here. 134 00:09:43,760 --> 00:09:49,670 Now though with the multiplication we can clearly see that 100 is going to cancel with 100 We have 1 135 00:09:49,670 --> 00:09:55,460 in the denominator and 1 in the numerator and then we can say that our final answer the sum of the series 136 00:09:55,820 --> 00:09:59,840 is going to be 21 over ninety nine and we can leave it in fraction form. 137 00:09:59,990 --> 00:10:07,100 Or we could use our calculators to find a decimal representation of 21 over 99 but 21 over 99 is going 138 00:10:07,100 --> 00:10:10,330 to be the sum of this convergent geometric series. 15600