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These are the user uploaded subtitles that are being translated: 1 00:00:00,370 --> 00:00:04,730 In this video we're talking about the difference between the limit of a series and the sum of a series. 2 00:00:04,800 --> 00:00:10,560 And in this particular problem we have the infinite series from equals one to infinity of 4000 96 race 3 00:00:10,560 --> 00:00:15,470 to the end over to power divided by a race to the two and power. 4 00:00:15,630 --> 00:00:20,910 So when we're taking the limit of an infinite series we're interested in the value that the series approaches 5 00:00:21,210 --> 00:00:23,840 as and becomes very very very large. 6 00:00:23,850 --> 00:00:30,540 So the way that we would calculate that with calculus is we would say the limit as and approaches infinity 7 00:00:30,840 --> 00:00:37,100 of our series here so 40 96 to the and over to divided by eight to the two. 8 00:00:37,170 --> 00:00:41,450 And then this just becomes a limit problem where we have to evaluate this limit. 9 00:00:41,490 --> 00:00:46,170 So depending on the function you may have to evaluate the limit differently you might just be able to 10 00:00:46,170 --> 00:00:51,210 use a simple substitution to evaluate the limit you might have to use factoring or conjugate method. 11 00:00:51,210 --> 00:00:56,910 You might have to use those battels rule if you get an indeterminate form when you plug in this value 12 00:00:56,910 --> 00:00:58,110 and equals infinity. 13 00:00:58,170 --> 00:01:03,680 But in this case what we're going to be able to do is actually simplify the fraction itself. 14 00:01:03,690 --> 00:01:08,820 So what we see here is that we have the exponent and over 2 in the numerator. 15 00:01:08,910 --> 00:01:15,390 Well that's the same thing as saying four thousand ninety six to one half we take that two out of the 16 00:01:15,390 --> 00:01:17,820 denominator we put it in the dominator here. 17 00:01:17,970 --> 00:01:23,700 So essentially factor out the two from the denominator of 40 96 to the one half all race to the end 18 00:01:23,700 --> 00:01:29,040 power because then when we multiplied x points back together we get an over 2 which is our original 19 00:01:29,040 --> 00:01:29,660 exponent. 20 00:01:29,760 --> 00:01:36,520 And of course that's just the same as the square root of 40 96 all raised to the power. 21 00:01:36,600 --> 00:01:40,010 The square root of 40 96 is 64. 22 00:01:40,050 --> 00:01:47,040 So we end up with then is 64 to the power in the numerator in the denominator here we can think about 23 00:01:47,040 --> 00:01:54,330 this as 8 squared and then raised to the power because when we multiply two times and we get two when 24 00:01:54,330 --> 00:01:56,080 we're back to our original exponent. 25 00:01:56,130 --> 00:01:57,880 So we just factored out that too. 26 00:01:58,020 --> 00:02:00,680 Well of course eight squared is 64. 27 00:02:00,720 --> 00:02:04,390 So we end up with is just 64 to the power. 28 00:02:04,410 --> 00:02:12,980 So now we're talking about the limit as and approaches infinity of 64 and divided by 64 n. 29 00:02:13,080 --> 00:02:16,970 Well of course one thing divided by itself is just one. 30 00:02:17,100 --> 00:02:24,250 So the value of this fraction is 1 so we get the limit as and approaches infinity of 1. 31 00:02:24,450 --> 00:02:27,000 Well we have no variable to plug in for here. 32 00:02:27,030 --> 00:02:32,420 So no matter what the value is of n the limit of the function is going to be 1. 33 00:02:32,520 --> 00:02:37,980 So we can say then that the limit is just equal to 1 because the limit never changes regardless of the 34 00:02:37,980 --> 00:02:40,590 value of N because there's no variable in this function. 35 00:02:40,590 --> 00:02:42,630 So the limit is always equal to 1. 36 00:02:42,690 --> 00:02:49,610 What that means if we were to graph this series so let's put a graph together really quickly up here. 37 00:02:49,890 --> 00:02:56,370 So if we were to graph this series what it would show us is that the function looks like this. 38 00:02:56,390 --> 00:03:04,260 If this is why equals 1 so this is one here than the limit or the value of this series is always going 39 00:03:04,260 --> 00:03:08,520 to be 1 because the function simplifies to 1. 40 00:03:08,520 --> 00:03:11,000 So essentially our series here is one. 41 00:03:11,040 --> 00:03:13,620 Therefore the limit of the series is 1. 42 00:03:13,620 --> 00:03:17,190 Now what happens when we want to find the sum of the series. 43 00:03:17,200 --> 00:03:20,030 Well remember our series simplified to 1. 44 00:03:20,040 --> 00:03:26,870 So we would have the sum from unequals 1 to infinity of 1 because 40 96 to the end over 2. 45 00:03:26,910 --> 00:03:29,560 Divided by eight to the two N is equal to 1. 46 00:03:29,580 --> 00:03:30,920 So we'd have the same here. 47 00:03:30,930 --> 00:03:36,480 So then if we plugged unequals 1 into our function we start trying to find the sum here we start expanding 48 00:03:36,480 --> 00:03:37,040 the sum. 49 00:03:37,110 --> 00:03:40,060 We plug in any one where there's no variable to plug in for. 50 00:03:40,060 --> 00:03:43,240 We so just get one if we plug in N equals 2. 51 00:03:43,260 --> 00:03:49,860 We still just get once we do I wonder that if we plug n equals 3 into our series we still just get one. 52 00:03:49,950 --> 00:03:52,570 So we'd add 1 to that and we would keep going. 53 00:03:52,680 --> 00:03:55,860 We would keep adding one on into infinity. 54 00:03:55,890 --> 00:04:01,950 And so if we keep adding 1 to the sum then obviously this sum is going to be infinite and the sum of 55 00:04:01,950 --> 00:04:03,780 the series is infinity. 56 00:04:03,780 --> 00:04:05,160 Another way you could think about that. 57 00:04:05,190 --> 00:04:12,300 If you just plugged unequals one two three four into this original series here or even into 64 to the 58 00:04:12,300 --> 00:04:14,070 end over 64 to the end. 59 00:04:14,190 --> 00:04:20,520 Obviously we would have 64 to the first over 64 to the first plus 64 to the second. 60 00:04:20,550 --> 00:04:26,880 Over 64 to the second plus 64 to the third over 64 to the third. 61 00:04:27,000 --> 00:04:32,220 And we could go on like that forever this would be the term that corresponds to an equals 1 and equals 62 00:04:32,220 --> 00:04:33,770 to an end equals three. 63 00:04:33,780 --> 00:04:38,680 But obviously here we would just get one plus one plus one. 64 00:04:38,790 --> 00:04:41,760 And so we would have a sum of infinity. 65 00:04:41,790 --> 00:04:47,130 So you can see here that the limit of the series is one we can set a limit of the series. 66 00:04:47,130 --> 00:04:49,460 Here is one. 67 00:04:49,770 --> 00:04:53,490 But the sum of the series is infinity. 68 00:04:53,490 --> 00:04:58,260 And so you can see that we get different values for the limit and the sum of the series needed to make 69 00:04:58,260 --> 00:04:59,730 sure you know how to find both. 7508