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These are the user uploaded subtitles that are being translated: 1 00:00:00,570 --> 00:00:04,800 Today we're going to be talking about how to find the power series representation of a function and 2 00:00:04,800 --> 00:00:08,840 the associated radius an interval of convergence for the power series. 3 00:00:09,000 --> 00:00:13,860 And then this particular problem we've been given the function f of x is equal to x divided by the quantity 4 00:00:14,170 --> 00:00:16,610 9 plus x squared. 5 00:00:16,650 --> 00:00:21,120 So the first thing we need to do is find a power series representation of this function and the way 6 00:00:21,120 --> 00:00:26,820 that we're going to do it is by comparing it to this well-known power series The Power series acts to 7 00:00:26,820 --> 00:00:27,800 the power. 8 00:00:27,990 --> 00:00:33,870 We know that the sum of that power series is one divided by the quantity one minus X so what we're going 9 00:00:33,870 --> 00:00:41,250 to do is compare our original function r f of x function here to the sum one divided by 1 minus X because 10 00:00:41,250 --> 00:00:42,540 they're already somewhat similar. 11 00:00:42,540 --> 00:00:47,610 They're both rational functions we've got two terms here in the denominator of each one term in the 12 00:00:47,700 --> 00:00:48,930 numerator of each. 13 00:00:49,080 --> 00:00:50,560 We're going to compare those two together. 14 00:00:50,580 --> 00:00:58,170 Try to make our function f of x look more similar to this some here 1 divided by 1 minus X and in that 15 00:00:58,170 --> 00:01:02,980 way hope to find a power series representation for our function f of x. 16 00:01:03,030 --> 00:01:07,710 So in order to make our function look more like this some one divided by 1 minus x. 17 00:01:07,770 --> 00:01:12,990 The first thing we want to do is factor an X out of the numerator of our original function. 18 00:01:12,990 --> 00:01:22,290 So if we take an x out and we say that this is X times 1 divided by 9 plus x squared we haven't changed 19 00:01:22,320 --> 00:01:25,380 the function at all we just factor the X out of the numerator. 20 00:01:25,380 --> 00:01:30,990 But now what we have inside the parentheses here we have one in the numerator just like we have 1 in 21 00:01:30,990 --> 00:01:34,050 the numerator of this some 1 over 1 minus. 22 00:01:34,080 --> 00:01:36,290 So we're already closer. 23 00:01:36,330 --> 00:01:42,060 Now what we want to try to do is get a one a value of 1 for the first term of our denominator and the 24 00:01:42,060 --> 00:01:46,470 way we're going to do that is by factoring out and 9 from our denominator. 25 00:01:46,470 --> 00:01:52,320 So this is going to be come here now x times and then inside the parentheses we'll still have 1 in our 26 00:01:52,320 --> 00:01:53,050 numerator. 27 00:01:53,280 --> 00:01:55,420 But in our denominator will factor out a 9. 28 00:01:55,440 --> 00:01:58,660 And we'll get nine times one plus. 29 00:01:58,710 --> 00:02:05,720 Now in order to make this still true what we have to do is call this x squared divided by nine. 30 00:02:05,730 --> 00:02:11,130 That way when we multiply x squared divided by nine times nine are nine cancel and we're just left with 31 00:02:11,190 --> 00:02:15,470 x squared which is what we had originally here just this x squared term. 32 00:02:15,480 --> 00:02:18,280 So that's the way that we keep that still true. 33 00:02:18,330 --> 00:02:25,710 But now if we factor this nine outside of the parentheses what you can see we have is X divided by nine 34 00:02:25,770 --> 00:02:34,610 times what's inside of parentheses here 1 over 1 plus x squared divided by 9 like that. 35 00:02:34,620 --> 00:02:36,410 Now we're really close to this. 36 00:02:36,420 --> 00:02:40,660 We have one in the numerator like are some over here 1 over 1 minus x. 37 00:02:40,670 --> 00:02:43,650 We have our first term in the denominator as a 1. 38 00:02:43,650 --> 00:02:48,660 Just like this some over here the only difference now is the second term and the denominator and the 39 00:02:48,660 --> 00:02:54,900 fact that this sum has a negative sign here and the denominator verses are sum which has a positive 40 00:02:54,900 --> 00:02:57,330 sign here in the middle of our denominator. 41 00:02:57,330 --> 00:03:02,040 So what we're going to do is instead of this positive sign here we're going to turn this into a double 42 00:03:02,040 --> 00:03:09,770 negative we're going to say that this is equal to x over nine times 1 divided by 1 minus. 43 00:03:09,780 --> 00:03:15,620 And now we're going to call this negative x squared over 9 like this. 44 00:03:15,630 --> 00:03:18,480 What that does for us is it has this negative sign here. 45 00:03:18,480 --> 00:03:25,620 Now what we can do is say that this x value right here the x value in this particular spot in this sum 46 00:03:26,040 --> 00:03:31,290 is equal to this x value that we have here the x value in the same spot. 47 00:03:31,470 --> 00:03:35,800 And if we just say that x is equal to negative x squared divided by 9. 48 00:03:36,030 --> 00:03:37,710 Everything else is the same. 49 00:03:37,710 --> 00:03:44,340 So we can substitute this negative x squared plus 9 in for x right here because in this sum this infinite 50 00:03:44,340 --> 00:03:45,460 sum here. 51 00:03:45,600 --> 00:03:49,220 Pulling this x value from this sum right here this x value. 52 00:03:49,380 --> 00:03:54,140 So when ours is negative x squared over 9 we just plug that in right there. 53 00:03:54,180 --> 00:03:56,980 We multiply by x divided by 9. 54 00:03:57,000 --> 00:03:59,340 And now we have a power series representation. 55 00:03:59,370 --> 00:04:06,090 So our power series representation is going to be the sum from N equals zero to infinity and we're taking 56 00:04:06,090 --> 00:04:09,920 this directly from the well-known power series here. 57 00:04:10,140 --> 00:04:15,750 And then for X we're substituting negative x squared divided by nine so we're going to say negative 58 00:04:16,200 --> 00:04:18,020 x squared divided by nine. 59 00:04:18,270 --> 00:04:22,530 That's all raised to the end power according to our well-known power series here. 60 00:04:22,860 --> 00:04:29,500 So now we've represented this part right here this one over one minus X which we had right here inside 61 00:04:29,500 --> 00:04:31,920 our parentheses but we haven't accounted for this. 62 00:04:31,920 --> 00:04:33,460 X divided by 9. 63 00:04:33,480 --> 00:04:38,430 So we have to multiply this by X overnight and we'll just put that out in front. 64 00:04:38,490 --> 00:04:40,800 So this is our power series representation. 65 00:04:40,800 --> 00:04:42,950 All we have to do is simplify it. 66 00:04:42,960 --> 00:04:48,210 And the way that we're going to do that is by bringing the X divided by nine inside the infinite sum 67 00:04:48,210 --> 00:04:54,850 here so we're going to say that this is going to be the some from an equals zero to infinity. 68 00:04:54,870 --> 00:04:56,940 We're going to bring the X divided by nine inside. 69 00:04:56,940 --> 00:05:01,820 We're going to basically have X to the first power divided by 9:00 to the first power right both of 70 00:05:01,820 --> 00:05:06,950 these are raised to the first power and then here for our negative x squared divided by nine all race 71 00:05:06,950 --> 00:05:08,050 to the end power. 72 00:05:08,210 --> 00:05:13,070 The first thing we want to do with that is pull out the negative one value we can pull out this negative 73 00:05:13,070 --> 00:05:19,450 here as a negative one race to the end whenever we have a negative value inside parentheses like that. 74 00:05:19,580 --> 00:05:20,580 We can pull it out. 75 00:05:20,630 --> 00:05:25,260 We just want to make sure it's raised to this xponent here so we have negative one to the end. 76 00:05:25,430 --> 00:05:30,800 What that leaves us with is just everything left inside our parentheses x squared divided by nine. 77 00:05:30,830 --> 00:05:33,670 So we get X squared divided by nine. 78 00:05:33,680 --> 00:05:38,390 We just need to make sure that we keep it raised to the power so to the end power like that. 79 00:05:38,390 --> 00:05:44,510 Now if we simplify further we'll say we have some from an equal zero to infinity will bring the negative 80 00:05:44,510 --> 00:05:46,550 one to the an out in front. 81 00:05:46,550 --> 00:05:49,850 That's always a good idea a negative one to the power. 82 00:05:49,850 --> 00:05:53,660 Now notice here we have x rays to the to race to the nth power. 83 00:05:53,670 --> 00:05:58,280 Whenever you have two exponents like that something raised to an exponent raise to another xponent you 84 00:05:58,280 --> 00:06:00,050 can multiply those exponent together. 85 00:06:00,050 --> 00:06:03,420 So this is going to become X to the to end power. 86 00:06:03,500 --> 00:06:08,630 Same thing here you basically have nine to the first power race to the end you multiply one times and 87 00:06:08,850 --> 00:06:09,700 and you get in. 88 00:06:09,740 --> 00:06:12,620 So this is nine to the end instead of nine. 89 00:06:12,620 --> 00:06:14,250 The first race to the end. 90 00:06:14,330 --> 00:06:18,220 So now we have things with like bases that we can combine. 91 00:06:18,230 --> 00:06:23,930 So we have here to the first power and we have X to the two and power. 92 00:06:23,990 --> 00:06:27,620 Whenever you have two terms with like Bass's they both have a base of x. 93 00:06:27,680 --> 00:06:30,620 You can combine them as long as you add the exponent together. 94 00:06:30,620 --> 00:06:37,730 So we have an exponent here to N and an exponent 1 so we can combine these and say X to the two and 95 00:06:37,790 --> 00:06:39,910 plus one we just add them together. 96 00:06:40,040 --> 00:06:47,140 Same thing here with our denominator we have nine to the first power and we have nine to the end power. 97 00:06:47,270 --> 00:06:53,450 So we can combine them and we can say this is divided by nine to the end plus one we just add those 98 00:06:53,450 --> 00:06:54,680 exponents together. 99 00:06:54,680 --> 00:06:56,980 Now we've simplified this as much as we can. 100 00:06:56,990 --> 00:07:01,610 This is our power series representation of the original function f of x. 101 00:07:01,610 --> 00:07:04,460 We've got the first half of the problem done. 102 00:07:04,460 --> 00:07:10,810 Now we need to do is find the associated radius an interval of convergence for this power series. 103 00:07:10,880 --> 00:07:15,980 The easiest way to do that is to find the radius of convergence first and then use that radius to find 104 00:07:15,980 --> 00:07:17,970 the interval of convergence. 105 00:07:18,040 --> 00:07:22,970 The way that we're going to find the radius of convergence is we're going to use the ratio test and 106 00:07:22,970 --> 00:07:26,050 what the ratio test tells us we're going to call this L. 107 00:07:26,150 --> 00:07:34,910 We're going to set that equal to the limit as an goes to infinity of the absolute value of a seven plus 108 00:07:34,910 --> 00:07:37,520 one divided by a seven. 109 00:07:37,520 --> 00:07:39,460 This is the ratio test right here. 110 00:07:39,740 --> 00:07:45,470 What it tells you is that if you evaluate this right hand side based on your power series up here and 111 00:07:45,470 --> 00:07:51,540 you find a value for L you get some value of L if you set that value of L to be less than 1. 112 00:07:51,710 --> 00:07:54,930 That tells you where your series will converge. 113 00:07:54,950 --> 00:08:01,870 So this is an extremely useful convergence test for us to use to find the radius of convergence. 114 00:08:02,000 --> 00:08:03,070 All we need to do. 115 00:08:03,140 --> 00:08:10,460 We're going to say L is equal to the limit as and goes to infinity of the absolute value here for a 116 00:08:10,460 --> 00:08:11,680 seven plus one. 117 00:08:11,780 --> 00:08:16,350 We just plug and plus one into our original power series here. 118 00:08:16,370 --> 00:08:18,040 Every hour we have N. 119 00:08:18,080 --> 00:08:23,990 So we say negative 1 raise 2 instead of the n and plus 1. 120 00:08:24,110 --> 00:08:29,240 Then we multiply that by x to the two times the quantity and plus one plus one. 121 00:08:29,240 --> 00:08:31,300 So here we're going to get 2 times. 122 00:08:31,400 --> 00:08:34,080 And plus one plus one. 123 00:08:34,190 --> 00:08:34,740 When we play. 124 00:08:34,760 --> 00:08:37,070 And plus 1 and for this value right here. 125 00:08:37,070 --> 00:08:42,440 We get two and plus two plus one or two and plus three. 126 00:08:42,440 --> 00:08:45,810 So we have X to the two and plus three. 127 00:08:45,950 --> 00:08:54,800 Then we divide that by nine to the end plus 1 plus 1 or 9 to the end plus 2. 128 00:08:55,040 --> 00:08:58,980 That's a sub and plus one right here that we got. 129 00:08:59,150 --> 00:09:04,880 We divide that then by a sub N which is just our original series here. 130 00:09:04,880 --> 00:09:08,330 This whole thing right here we divide that by this part right here. 131 00:09:08,390 --> 00:09:14,360 But because we have a fraction divided by a fraction instead of just dividing we can kind of combine 132 00:09:14,360 --> 00:09:15,370 two steps here. 133 00:09:15,470 --> 00:09:19,330 And instead of dividing by this value we can multiply by the reciprocal. 134 00:09:19,330 --> 00:09:24,000 So then instead of a big division problem we get a multiplication problem instead. 135 00:09:24,050 --> 00:09:28,790 So we just take their supercool in other words we flip this upside down our denominator 9 to the end 136 00:09:28,790 --> 00:09:35,900 plus one becomes our numerator nine to the end plus 1 and our numerator negative 1 to the n negative 137 00:09:35,900 --> 00:09:43,150 1 to the end times x to the two and plus 1 which was our numerator becomes our denominator. 138 00:09:43,190 --> 00:09:48,020 This is the limit that we're going to be evaluating from here we just need to combine like terms so 139 00:09:48,020 --> 00:09:54,980 we're looking for things with like Bass's we're going to say L is equal to the limit as and goes to 140 00:09:55,100 --> 00:09:57,320 infinity of the absolute value. 141 00:09:57,320 --> 00:10:01,680 Now from here we're looking for like bases like a so we have this value here. 142 00:10:01,690 --> 00:10:07,840 Negative One race to the end plus one in our numerator and we have negative one race to the end in our 143 00:10:07,840 --> 00:10:12,210 denominator because we have like Bass's we can consolidate these terms. 144 00:10:12,280 --> 00:10:17,510 We just need to subtract the exponent in the denominator from the exponent in the numerator so the exponent 145 00:10:17,530 --> 00:10:20,500 in our numerator is this and plus one right here. 146 00:10:20,770 --> 00:10:25,800 So we're going to get an A plus one from the numerator the exponent and the nominator is n. 147 00:10:25,900 --> 00:10:32,010 So we're going to subtract that X point in the denominator so minus an and plus 1 minus. 148 00:10:32,020 --> 00:10:33,490 And our ends are going to cancel. 149 00:10:33,490 --> 00:10:36,240 And we're just going to be left with positive 1. 150 00:10:36,250 --> 00:10:42,210 So we're going to get negative 1 race to the positive one power in our numerator or just negative ones 151 00:10:42,210 --> 00:10:45,270 so we'll get a negative sign here to represent that in our numerator. 152 00:10:45,310 --> 00:10:51,970 When we take X to the two n plus three minus X to the two n minus 1 we subtract those exponents are 153 00:10:52,000 --> 00:10:53,590 two ends are going to cancel. 154 00:10:53,620 --> 00:10:55,850 We're going to get three minus one which is just two. 155 00:10:55,990 --> 00:11:02,500 We're going to be left with positive x squared here in our numerator then same thing here 9 to the end 156 00:11:02,500 --> 00:11:05,380 plus 1 over at to the end plus 2. 157 00:11:05,440 --> 00:11:10,640 When we subtract the exponent and the denominator from the exponent in the numerator our ends will cancel. 158 00:11:10,710 --> 00:11:14,270 We'll be left with 1 minus 2 or just negative 1. 159 00:11:14,290 --> 00:11:16,160 So we have nine to the negative 1. 160 00:11:16,210 --> 00:11:21,010 We know because we have a negative exponent that that value is going to be in a denominator and we have 161 00:11:21,100 --> 00:11:24,270 then 9 to the positive one in the denominator. 162 00:11:24,280 --> 00:11:30,550 So this is our simplified absolute value here now because we were able to cancel all of our values out 163 00:11:30,550 --> 00:11:31,820 of this function. 164 00:11:31,870 --> 00:11:35,270 This limit as and goes to infinity becomes irrelevant. 165 00:11:35,300 --> 00:11:41,590 It's only relevant if we have an end value left over but we don't have any values of x values so this 166 00:11:41,680 --> 00:11:48,130 limit is n goes to infinity really just goes away and what we are left with is that L is equal to the 167 00:11:48,130 --> 00:11:51,610 absolute value of negative x squared divided by 9. 168 00:11:51,760 --> 00:11:56,380 Because we have these absolute value brackets we can cancel out this positive sign and just say the 169 00:11:56,380 --> 00:11:59,620 absolute value of x squared over 9. 170 00:11:59,620 --> 00:12:04,900 Remember that we said at the beginning when we started this ratio test process that once we found a 171 00:12:04,900 --> 00:12:10,940 value for L we would set it less than 1 and that would give us our radius of convergence. 172 00:12:11,050 --> 00:12:12,840 And it does give us our radius of convergence. 173 00:12:12,850 --> 00:12:19,370 We just need to solve this inequality for a value of x and right now we have x squared divided by 9. 174 00:12:19,390 --> 00:12:26,350 So we're going to do is multiply both sides by nine and we'll be left with the absolute value of x squared. 175 00:12:26,350 --> 00:12:32,050 Less than nine we get nine times 1 which is nine over here in order to get rid of our absolute value 176 00:12:32,050 --> 00:12:36,920 bars and take the square root of this x squared value just to get X on its own. 177 00:12:37,150 --> 00:12:44,230 We need to say that x is greater than negative 3 and less than positive 3 right when we take the square 178 00:12:44,230 --> 00:12:50,890 root of x squared we get x we get positive 3 over here taking the square means we can also get negative 179 00:12:50,890 --> 00:12:51,220 3. 180 00:12:51,230 --> 00:12:54,280 We have to say x is greater than negative 3. 181 00:12:54,400 --> 00:12:59,440 This value now that we just found this X less than 3 right here. 182 00:12:59,440 --> 00:13:06,940 This gives us our radius of convergence it tells us that our radius of convergence is our equals 3 because 183 00:13:06,940 --> 00:13:11,380 once we solve for x whatever value we get right here is our radius of convergence. 184 00:13:11,410 --> 00:13:17,430 So now our radius of convergence is three we can say that our interval of convergence is negative 3 185 00:13:17,470 --> 00:13:21,630 to positive 3 we just take both sides of this inequality here. 186 00:13:21,940 --> 00:13:28,300 And this is fine except that we have to test the endpoints of the interval of convergence to see whether 187 00:13:28,300 --> 00:13:33,690 or not the value negative 3 itself is included in the interval of convergence and whether or not the 188 00:13:33,690 --> 00:13:40,240 value positive 3 itself is included in the interval of convergence or if the values between them are 189 00:13:40,240 --> 00:13:45,460 the only values for which the series converges and the end points actually aren't included in order 190 00:13:45,460 --> 00:13:48,470 to test the endpoints of our interval of convergence. 191 00:13:48,490 --> 00:13:54,540 We just want to plug the endpoints into the power series representation that we found earlier our function. 192 00:13:54,610 --> 00:13:56,970 We're going to be plugging in those end points for x. 193 00:13:56,980 --> 00:13:59,030 Not for N but for x. 194 00:13:59,050 --> 00:14:06,450 So let's start with negative 3 let's plug negative 3 in for x so with X equals negative 3. 195 00:14:06,550 --> 00:14:14,740 Here's what we're going to get the some from and equals zero to infinity of negative 1 to the nth power 196 00:14:14,820 --> 00:14:23,860 usually plug in negative 3 so you get negative 3 raised to the two and plus 1 divided by 9 to the end 197 00:14:23,860 --> 00:14:24,850 plus 1. 198 00:14:25,090 --> 00:14:29,880 And really this is just going to test our skills with algebra and exponents. 199 00:14:29,890 --> 00:14:36,040 We just need to simplify this function and determine whether or not this new series right here converges 200 00:14:36,100 --> 00:14:42,970 or diverges and we can use any convergence test that we want to figure out whether or not this series 201 00:14:42,970 --> 00:14:44,470 converges or diverges. 202 00:14:44,470 --> 00:14:47,820 But before we do that we want to simplify it as much as we can. 203 00:14:47,830 --> 00:14:52,510 So in order to simplify it the first thing you need to realize is that we can reverse the process we 204 00:14:52,510 --> 00:14:54,500 used earlier with exponents. 205 00:14:54,580 --> 00:15:00,130 Remember that we said when we had it terms with like Bass's we can combine them by just the exponents 206 00:15:00,130 --> 00:15:06,520 together well we can separate them by just separating the exponent so here we have negative 3 race to 207 00:15:06,520 --> 00:15:14,320 the two and plus one we can separate these and say negative three to the two N times negative 3 to the 208 00:15:14,320 --> 00:15:15,210 first power. 209 00:15:15,220 --> 00:15:19,970 Here we can do the same thing in the denominator we have nine to the end plus 1. 210 00:15:20,170 --> 00:15:27,790 We can call this nine to the N times 9 to the first power so times nine to the first power like this 211 00:15:27,940 --> 00:15:31,620 similar thing here we have negative three raised to the two and power. 212 00:15:31,630 --> 00:15:36,190 Remember how we said if we had some base here raised to an exponent raise to another exponent we could 213 00:15:36,190 --> 00:15:38,050 multiply those x points together. 214 00:15:38,200 --> 00:15:40,860 Well here we have the product of two xponent two. 215 00:15:40,900 --> 00:15:46,930 And and we can separate those and instead of saying negative three to the two n we can say negative 216 00:15:46,930 --> 00:15:52,830 three squared raised to the power and we probably put parentheses around this. 217 00:15:52,840 --> 00:15:57,430 But now essentially what we have is negative 3 squared which is 9 raised to the power. 218 00:15:57,430 --> 00:16:03,700 So this is going to become nine to the positive and from here we can cancel We have nine to the end 219 00:16:03,700 --> 00:16:06,900 in the denominator and 9 to the end in the numerator. 220 00:16:06,910 --> 00:16:09,220 So those all go away like this. 221 00:16:09,220 --> 00:16:15,580 And what we're left with is really just the sum from an equals zero to infinity. 222 00:16:15,580 --> 00:16:20,620 Here we have negative three to the first and nine to the first power so we're just left with negative 223 00:16:20,620 --> 00:16:26,820 3 over 9 or negative one third times negative 1 to the n. 224 00:16:26,950 --> 00:16:32,110 What we should be able to see here is that this is just a geometric series remember geometric series 225 00:16:32,410 --> 00:16:37,270 comes in the form a times x rays to the end. 226 00:16:37,360 --> 00:16:44,230 Power will here are value of a is negative one third that coefficient r x value is this negative one 227 00:16:44,230 --> 00:16:45,800 race to the end power. 228 00:16:45,850 --> 00:16:51,550 Remember that the geometric series Test says that the series converges if the absolute value of x is 229 00:16:51,640 --> 00:16:52,750 less than 1. 230 00:16:52,840 --> 00:16:55,210 What are x value is this negative one right here. 231 00:16:55,210 --> 00:16:59,460 So we say the absolute value of negative 1 less than 1. 232 00:16:59,470 --> 00:17:03,270 We take the absolute value of this we get positive 1 less than 1. 233 00:17:03,280 --> 00:17:04,820 Well this is not true. 234 00:17:04,960 --> 00:17:12,040 So what we know is that at the end point negative 3 the series diverges by the geometric series test. 235 00:17:12,040 --> 00:17:19,320 So we're just going to say for this endpoint the series diverges by geometric series test. 236 00:17:19,360 --> 00:17:25,060 So what that means because the series diverges at this endpoint it means that we leave this interval 237 00:17:25,060 --> 00:17:28,320 of convergence with a parentheses on this left side here. 238 00:17:28,330 --> 00:17:34,570 If we found that the series converged at this negative 3 and point then we would replace this parentheses 239 00:17:34,840 --> 00:17:37,590 with a hard bracket like this. 240 00:17:37,600 --> 00:17:42,070 But because it diverges we just leave this parentheses here. 241 00:17:42,070 --> 00:17:45,060 Now we have to test the other end point positive three. 242 00:17:45,160 --> 00:17:48,250 So we say at X equals positive 3. 243 00:17:48,430 --> 00:17:49,180 What do we get. 244 00:17:49,210 --> 00:17:56,140 Well we have the sum from N equals zero to infinity of negative 1 to the power. 245 00:17:56,200 --> 00:18:03,250 Here's where we plug in our positive 3 so positive 3 goes in here for x positive 3 rays to the two and 246 00:18:03,280 --> 00:18:09,970 plus 1 divided by 9 to the end plus 1 and we have the same process here with algebra. 247 00:18:09,970 --> 00:18:13,390 We're going to call our denominator instead of 9 to the end plus 1. 248 00:18:13,390 --> 00:18:21,880 We're going to call it 9 to the nth power times 9 to the first power like this in our numerator instead 249 00:18:21,880 --> 00:18:29,560 of three to the two and plus 1 we're going to get three to the two N times 3 to the first power. 250 00:18:29,560 --> 00:18:35,830 We're going to instead of three to the two and we're going to call this three squared raised to the 251 00:18:35,980 --> 00:18:39,110 power and then instead of three squared we're going to call this nine. 252 00:18:39,110 --> 00:18:41,170 So this is going to become nine to the end. 253 00:18:41,410 --> 00:18:46,780 We're going to get nine to the end and nine to the end to cancel from our numerator and denominator 254 00:18:47,150 --> 00:18:54,010 and you can see that all we're left with is the sum from N equals zero to infinity negative 1 to the 255 00:18:54,010 --> 00:18:54,970 nth power. 256 00:18:54,970 --> 00:19:01,350 We're just left with three divided by nine or one third we can put that one third out in front there 257 00:19:01,360 --> 00:19:06,500 it's the same as what we had before except positive ONE-THIRD instead of negative one third. 258 00:19:06,520 --> 00:19:13,780 But the fact remains that this value here that we had for X because this is a geometric series is unchanged. 259 00:19:13,780 --> 00:19:16,600 We're going to get this same inequality here. 260 00:19:16,840 --> 00:19:18,820 Yes the value of X less than 1. 261 00:19:18,910 --> 00:19:23,770 We're going to plug in this negative 1 value for x and we're going to find that one is not less than 262 00:19:23,770 --> 00:19:24,100 1. 263 00:19:24,130 --> 00:19:25,500 They're equal to each other. 264 00:19:25,570 --> 00:19:32,690 And so by the same test we're going to say diverges by the geometric series test. 265 00:19:32,710 --> 00:19:38,620 So both end points the series diverges at both end points which means we leave the interval of convergence 266 00:19:38,710 --> 00:19:44,010 written this way with these sort of parentheses instead of hard brackets like this. 267 00:19:44,200 --> 00:19:47,260 So this is our interval of convergence. 268 00:19:47,260 --> 00:19:53,860 This is our radius of convergence and this is our power series representation here that we found for 269 00:19:53,860 --> 00:19:55,300 the function f of x. 30362