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These are the user uploaded subtitles that are being translated: 1 00:00:00,590 --> 00:00:06,440 Today we're talking about how to find the values for which the geometric series is convergent or converges. 2 00:00:06,540 --> 00:00:11,880 And in this particular problem we've been given the infinite sum from N equals 1 to infinity of the 3 00:00:11,880 --> 00:00:17,240 series negative five raised to the power times x rays to the power. 4 00:00:17,250 --> 00:00:22,440 Now we wouldn't know necessarily unless we were told that this was a geometric series so in order to 5 00:00:22,440 --> 00:00:27,930 prove first that it is a geometric series all we want to do is write out the first couple of terms of 6 00:00:27,930 --> 00:00:34,920 the series so that we can make sure that it is in fact in the form here that we have for a geometric 7 00:00:34,980 --> 00:00:42,060 series which is a times the quantity 1 plus Arpels are squared plus our cubed where r is that constant 8 00:00:42,120 --> 00:00:47,700 multiples so the first thing we want to do is recognize that in this particular problem this series 9 00:00:47,700 --> 00:00:53,090 that we've been given we have two values that are raised to the same exponent. 10 00:00:53,310 --> 00:00:58,220 When that's the case what we can do is actually combine the bases and just raise them both to the end 11 00:00:58,230 --> 00:00:58,630 power. 12 00:00:58,650 --> 00:01:07,350 So this series now becomes instead of these two separate terms here becomes negative 5 x all raised 13 00:01:07,410 --> 00:01:08,530 to the power. 14 00:01:08,700 --> 00:01:11,400 That simplifies our series A little bit. 15 00:01:11,400 --> 00:01:16,290 So now when it comes to writing out the first several terms of the series what we want to do is start 16 00:01:16,290 --> 00:01:18,960 plugging in an equals one equals two. 17 00:01:18,960 --> 00:01:24,380 The reason we start with N equals 1 is because we're told to write here because we have any equals one. 18 00:01:24,660 --> 00:01:32,280 So we start with N equals 1 and we get negative 5 x rays to the first power when we plug in an equals 19 00:01:32,280 --> 00:01:40,950 two we get plus negative 5 x raised to the power of 2 if we keep going we'll get negative 5 x to the 20 00:01:40,950 --> 00:01:49,210 third and then we'll get negative 5 x to the fourth etc. and we could continue on but you get the idea. 21 00:01:49,560 --> 00:01:52,940 So now can we get this series in the form here. 22 00:01:52,980 --> 00:01:55,020 Eight times one plus etc.. 23 00:01:55,080 --> 00:02:00,870 Well we recognize that we need what is inside the parentheses here to start with one. 24 00:02:00,930 --> 00:02:05,550 That means that in order to do that we need to factor out the first term entirely so knows this series 25 00:02:05,550 --> 00:02:08,370 here starts with a we factored out a. 26 00:02:08,370 --> 00:02:11,160 And that just left us with this series starting with 1. 27 00:02:11,220 --> 00:02:15,480 So we're going to factor out the first term entirely and we're going to say here we're going to factor 28 00:02:15,480 --> 00:02:18,010 out negative 5 x. 29 00:02:18,210 --> 00:02:24,060 And when we do that when we factor that out we're doing it negative 5 x times 1 because negative 5 x 30 00:02:24,060 --> 00:02:27,420 times 1 gives us this negative 5 x to the first power here. 31 00:02:27,750 --> 00:02:35,310 Then we're going to get plus negative 5 x because negative 5 x times negative 5 x gives us this quantity 32 00:02:35,340 --> 00:02:37,080 negative 5 x square. 33 00:02:37,080 --> 00:02:44,230 Then we're going to get plus negative 5 x squared because we have here negative 5 x cubed. 34 00:02:44,280 --> 00:02:49,710 We factored out one of them negative 5 x times negative 5 x squared gives us this quantity negative 35 00:02:49,710 --> 00:02:50,860 5 x cubed. 36 00:02:50,910 --> 00:02:52,870 So we could keep going here. 37 00:02:52,890 --> 00:02:56,830 Negative 5 x cubed Plus the dot dot. 38 00:02:56,910 --> 00:02:59,090 But you get the idea there. 39 00:02:59,250 --> 00:03:05,520 And what we see now is that we have we do in fact have a geometric series in the form that we've been 40 00:03:05,520 --> 00:03:06,090 given here. 41 00:03:06,090 --> 00:03:09,690 So we have this value here of a. 42 00:03:09,780 --> 00:03:16,830 In our case our value of a the value we factored out to get our first term was this negative 5 x. 43 00:03:16,830 --> 00:03:22,440 Now we have the rest of the series here starting with 1 and now we have a constant multipole this value 44 00:03:22,530 --> 00:03:26,310 of our value here of our. 45 00:03:26,400 --> 00:03:33,630 In our case that's this negative 5 x value and we can see that we do in fact for our next term have 46 00:03:33,720 --> 00:03:39,660 r squared and then r cubed r the fourth etc. which is the form we know that are geometric series is 47 00:03:39,660 --> 00:03:40,660 going to take. 48 00:03:40,710 --> 00:03:46,080 So we now know that we have a geometric series because we've matched to this forum and we know the value 49 00:03:46,170 --> 00:03:52,880 of a and are at this point we have everything we need to determine where the series is convergent. 50 00:03:52,890 --> 00:03:57,990 We know by the geometric series test that the series will be convergent where the absolute value of 51 00:03:58,050 --> 00:04:00,030 r is less than 1. 52 00:04:00,120 --> 00:04:05,250 The series is divergent where ever the absolute value of r is greater than or equal to one that's the 53 00:04:05,250 --> 00:04:07,750 geometric series test for convergence. 54 00:04:07,770 --> 00:04:11,700 We know that our value of r is negative 5 x. 55 00:04:11,880 --> 00:04:15,270 So that means that our series is going to be convergent. 56 00:04:15,470 --> 00:04:23,820 Wherever the absolute value of negative 5 x is less than 1 so now it's just a matter of solving this 57 00:04:23,820 --> 00:04:27,460 inequality to find out where our series is convergent. 58 00:04:27,510 --> 00:04:32,130 So the first thing we want to do is go ahead and take away this negative sign because these absolute 59 00:04:32,130 --> 00:04:36,620 value bars are going to get rid of this negative sign automatically for us anyway. 60 00:04:36,630 --> 00:04:41,070 So we're just left with the absolute value of 5 x is less than 1. 61 00:04:41,070 --> 00:04:48,240 Now in order to remove the absolute value bars we actually need to add to this inequality that 5 x has 62 00:04:48,240 --> 00:04:50,530 to be greater than negative 1. 63 00:04:50,550 --> 00:04:56,370 So negative 1 less than 5 x less than positive one. 64 00:04:56,490 --> 00:05:01,220 That's what we have to do when we take away the absolute value bars that we have to do it. 65 00:05:01,320 --> 00:05:05,690 And then in order to get a value for x we want to solve for a value for x. 66 00:05:05,820 --> 00:05:11,230 We just divide through each Pease's inequality by 5 to get x by itself. 67 00:05:11,400 --> 00:05:17,430 What we're left with here is negative one fifth less than X.. 68 00:05:17,610 --> 00:05:19,300 Less than positive. 69 00:05:19,320 --> 00:05:25,620 One fifth and we can see that the values for which the series is convergent are all values where X is 70 00:05:25,680 --> 00:05:29,340 between negative one fifth and positive one fifth. 71 00:05:29,340 --> 00:05:33,870 In other words greater than negative one fifth and same time less than positive one fifth. 72 00:05:33,870 --> 00:05:40,980 So within that interval the series is convergent outside that interval if X is less than negative one 73 00:05:40,980 --> 00:05:47,250 fifth or if x is greater than positive one fifth then we know on those intervals the series will be 74 00:05:47,280 --> 00:05:47,960 divergent. 75 00:05:47,970 --> 00:05:51,990 But within this small interval here the series is convergent. 8466