우선 문제 전문은
x=3에서 함수 f(x)=√x의 테일러 다항식을 이용하여, √8의 값을 소숫점 아래 4자리까지 정확하게 구하여라.
이구요 우선 테일러 다항식은
f(x)=√3+(x-3)/2√3+sigma n=2to∞ ((x^n)*((-1)^n))/(((2n-2)!)*(3^((2n-1)/2))*(2^n)))
로 구했구요
여기서 이제 나머지항을 통해서 몇항까지해야 정확하게 나오나 계산하는거까진 이해했는데요
저걸 엑셀로 돌려봤더니 100항넘게까지 합해봤는데 답도없더군요...
| 0 | 1 | 0.5 | 1 | 1 | 1.732050808 |
| 1 | 1 | -0.5 | 0.5 | 5 | 1.443375673 |
| 2 | 2 | -1.5 | -0.25 | 25 | -0.60140653 |
| 3 | 6 | -2.5 | 0.375 | 125 | 0.501172109 |
| 4 | 24 | -3.5 | -0.9375 | 625 | -0.52205428 |
| 5 | 120 | -4.5 | 3.28125 | 3125 | 0.609063327 |
| 6 | 720 | -5.5 | -14.7656 | 15625 | -0.761329158 |
| 7 | 5040 | -6.5 | 81.21094 | 78125 | 0.996978659 |
| 8 | 40320 | -7.5 | -527.871 | 390625 | -1.350075268 |
| 9 | 362880 | -8.5 | 3959.033 | 1953125 | 1.875104539 |
| 10 | 3628800 | -9.5 | -33651.8 | 9765625 | -2.656398097 |
| 11 | 39916800 | -10.5 | 319691.9 | 48828125 | 3.823603321 |
| 12 | 4.79E+08 | -11.5 | -3356765 | 2.44E+08 | -5.576088177 |
| 13 | 6.23E+09 | -12.5 | 38602801 | 1.22E+09 | 8.221155645 |
| 14 | 8.72E+10 | -13.5 | -4.8E+08 | 6.1E+09 | -12.23386257 |
| 15 | 1.31E+12 | -14.5 | 6.51E+09 | 3.05E+10 | 18.35079385 |
| 16 | 2.09E+13 | -15.5 | -9.4E+10 | 1.53E+11 | -27.71734488 |
| 17 | 3.56E+14 | -16.5 | 1.46E+12 | 7.63E+11 | 42.11949467 |
| 18 | 6.4E+15 | -17.5 | -2.4E+13 | 3.81E+12 | -64.34922797 |
| 19 | 1.22E+17 | -18.5 | 4.23E+14 | 1.91E+13 | 98.7817096 |
| 20 | 2.43E+18 | -19.5 | -7.8E+15 | 9.54E+13 | -152.288469 |
| 21 | 5.11E+19 | -20.5 | 1.53E+17 | 4.77E+14 | 235.6845353 |
| 22 | 1.12E+21 | -21.5 | -3.1E+18 | 2.38E+15 | -366.0252253 |
| 23 | 2.59E+22 | -22.5 | 6.72E+19 | 1.19E+16 | 570.2566916 |
| 24 | 6.2E+23 | -23.5 | -1.5E+21 | 5.96E+16 | -891.0260805 |
| 25 | 1.55E+25 | -24.5 | 3.55E+22 | 2.98E+17 | 1395.94086 |
| 26 | 4.03E+26 | -25.5 | -8.7E+23 | 1.49E+18 | -2192.343017 |
| 27 | 1.09E+28 | -26.5 | 2.22E+25 | 7.45E+18 | 3450.910304 |
| 28 | 3.05E+29 | -27.5 | -5.9E+26 | 3.73E+19 | -5443.400182 |
| 29 | 8.84E+30 | -28.5 | 1.62E+28 | 1.86E+20 | 8603.075 |
| 30 | 2.65E+32 | -29.5 | -4.6E+29 | 9.31E+20 | -13621.53542 |
| 31 | 8.22E+33 | -30.5 | 1.36E+31 | 4.66E+21 | 21604.04811 |
| 32 | 2.63E+35 | -31.5 | -4.1E+32 | 2.33E+22 | -34318.93059 |
| 33 | 8.68E+36 | -32.5 | 1.31E+34 | 1.16E+23 | 54598.29866 |
| 34 | 2.95E+38 | -33.5 | -4.2E+35 | 5.82E+23 | -86982.58365 |
| 35 | 1.03E+40 | -34.5 | 1.42E+37 | 2.91E+24 | 138757.9311 |
| 36 | 3.72E+41 | -35.5 | -4.9E+38 | 1.46E+25 | -221627.251 |
| 37 | 1.38E+43 | -36.5 | 1.74E+40 | 7.28E+25 | 354403.9374 |
| 38 | 5.23E+44 | -37.5 | -6.4E+41 | 3.64E+26 | -567357.1805 |
| 39 | 2.04E+46 | -38.5 | 2.39E+43 | 1.82E+27 | 909226.2508 |
| 40 | 8.16E+47 | -39.5 | -9.2E+44 | 9.09E+27 | -1458550.444 |
| 41 | 3.35E+49 | -40.5 | 3.63E+46 | 4.55E+28 | 2341981.404 |
| 42 | 1.41E+51 | -41.5 | -1.5E+48 | 2.27E+29 | -3763898.685 |
| 43 | 6.04E+52 | -42.5 | 6.1E+49 | 1.14E+30 | 6054333.156 |
| 44 | 2.66E+54 | -43.5 | -2.6E+51 | 5.68E+30 | -9746559.058 |
| 45 | 1.2E+56 | -44.5 | 1.13E+53 | 2.84E+31 | 15702789.59 |
| 46 | 5.5E+57 | -45.5 | -5E+54 | 1.42E+32 | -25317903.51 |
| 47 | 2.59E+59 | -46.5 | 2.28E+56 | 7.11E+32 | 40849808.86 |
| 48 | 1.24E+61 | -47.5 | -1.1E+58 | 3.55E+33 | -65955420.55 |
| 49 | 6.08E+62 | -48.5 | 5.04E+59 | 1.78E+34 | 106560628.4 |
| 50 | 3.04E+64 | -49.5 | -2.4E+61 | 8.88E+34 | -172273016 |
| 51 | 1.55E+66 | -50.5 | 1.21E+63 | 4.44E+35 | 278676937.6 |
| 52 | 8.07E+67 | -51.5 | -6.1E+64 | 2.22E+36 | -451063633 |
| 53 | 4.27E+69 | -52.5 | 3.15E+66 | 1.11E+37 | 730496135.2 |
| 54 | 2.31E+71 | -53.5 | -1.7E+68 | 5.55E+37 | -1183674293 |
| 55 | 1.27E+73 | -54.5 | 8.84E+69 | 2.78E+38 | 1918987112 |
| 56 | 7.11E+74 | -55.5 | -4.8E+71 | 1.39E+39 | -3112642785 |
| 57 | 4.05E+76 | -56.5 | 2.67E+73 | 6.94E+39 | 5051218555 |
| 58 | 2.35E+78 | -57.5 | -1.5E+75 | 3.47E+40 | -8200972654 |
| 59 | 1.39E+80 | -58.5 | 8.69E+76 | 1.73E+41 | 13320788915 |
| 60 | 8.32E+81 | -59.5 | -5.1E+78 | 8.67E+41 | -21646281987 |
| 61 | 5.08E+83 | -60.5 | 3.02E+80 | 4.34E+42 | 35189993940 |
| 62 | 3.15E+85 | -61.5 | -1.8E+82 | 2.17E+43 | -57231038531 |
| 63 | 1.98E+87 | -62.5 | 1.13E+84 | 1.08E+44 | 93113991261 |
| 64 | 1.27E+89 | -63.5 | -7E+85 | 5.42E+44 | -1.51553E+11 |
| 65 | 8.25E+90 | -64.5 | 4.47E+87 | 2.71E+45 | 2.46759E+11 |
| 66 | 5.44E+92 | -65.5 | -2.9E+89 | 1.36E+46 | -4.01918E+11 |
| 67 | 3.65E+94 | -66.5 | 1.89E+91 | 6.78E+46 | 6.54866E+11 |
| 68 | 2.48E+96 | -67.5 | -1.3E+93 | 3.39E+47 | -1.06737E+12 |
| 69 | 1.71E+98 | -68.5 | 8.47E+94 | 1.69E+48 | 1.74027E+12 |
| 70 | 1.2E+100 | -69.5 | -5.8E+96 | 8.47E+48 | -2.8383E+12 |
| 71 | 8.5E+101 | -70.5 | 4.03E+98 | 4.24E+49 | 4.63057E+12 |
| 72 | 6.1E+103 | -71.5 | -3E+100 | 2.12E+50 | -7.55683E+12 |
| 73 | 4.5E+105 | -72.5 | 2E+102 | 1.06E+51 | 1.23359E+13 |
| 74 | 3.3E+107 | -73.5 | -1E+104 | 5.29E+51 | -2.01431E+13 |
| 75 | 2.5E+109 | -74.5 | 1.1E+106 | 2.65E+52 | 3.29004E+13 |
| 76 | 1.9E+111 | -75.5 | -8E+107 | 1.32E+53 | -5.37518E+13 |
| 77 | 1.5E+113 | -76.5 | 6.1E+109 | 6.62E+53 | 8.78411E+13 |
| 78 | 1.1E+115 | -77.5 | -5E+111 | 3.31E+54 | -1.43586E+14 |
| 79 | 8.9E+116 | -78.5 | 3.6E+113 | 1.65E+55 | 2.34767E+14 |
| 80 | 7.2E+118 | -79.5 | -3E+115 | 8.27E+55 | -3.83941E+14 |
| 81 | 5.8E+120 | -80.5 | 2.3E+117 | 4.14E+56 | 6.28052E+14 |
| 82 | 4.8E+122 | -81.5 | -2E+119 | 2.07E+57 | -1.02761E+15 |
| 83 | 3.9E+124 | -82.5 | 1.5E+121 | 1.03E+58 | 1.68172E+15 |
| 84 | 3.3E+126 | -83.5 | -1E+123 | 5.17E+58 | -2.75282E+15 |
| 85 | 2.8E+128 | -84.5 | 1E+125 | 2.58E+59 | 4.50707E+15 |
| 86 | 2.4E+130 | -85.5 | -9E+126 | 1.29E+60 | -7.38077E+15 |
| 87 | 2.1E+132 | -86.5 | 7.4E+128 | 6.46E+60 | 1.20892E+16 |
| 88 | 1.9E+134 | -87.5 | -6E+130 | 3.23E+61 | -1.98052E+16 |
| 89 | 1.7E+136 | -88.5 | 5.6E+132 | 1.62E+62 | 3.24524E+16 |
| 90 | 1.5E+138 | -89.5 | -5E+134 | 8.08E+62 | -5.31858E+16 |
| 91 | 1.4E+140 | -90.5 | 4.4E+136 | 4.04E+63 | 8.71818E+16 |
| 92 | 1.2E+142 | -91.5 | -4E+138 | 2.02E+64 | -1.42934E+17 |
| 93 | 1.2E+144 | -92.5 | 3.7E+140 | 1.01E+65 | 2.34381E+17 |
| 94 | 1.1E+146 | -93.5 | -3E+142 | 5.05E+65 | -3.84402E+17 |
| 95 | 1E+148 | -94.5 | 3.2E+144 | 2.52E+66 | 6.30553E+17 |
| 96 | 9.9E+149 | -95.5 | -3E+146 | 1.26E+67 | -1.0345E+18 |
| 97 | 9.6E+151 | -96.5 | 2.9E+148 | 6.31E+67 | 1.69751E+18 |
| 98 | 9.4E+153 | -97.5 | -3E+150 | 3.16E+68 | -2.78587E+18 |
| 99 | 9.3E+155 | -98.5 | 2.7E+152 | 1.58E+69 | 4.57277E+18 |
| 100 | 9.3E+157 | -99.5 | -3E+154 | 7.89E+69 | -7.50697E+18 |
| 101 | 9.4E+159 | -100.5 | 2.6E+156 | 3.94E+70 | 1.23258E+19 |
| 102 | 9.6E+161 | -101.5 | -3E+158 | 1.97E+71 | -2.02409E+19 |
| 103 | 9.9E+163 | -102.5 | 2.7E+160 | 9.86E+71 | 3.32435E+19 |
| 104 | 1E+166 | -103.5 | -3E+162 | 4.93E+72 | -5.46068E+19 |
| 105 | 1.1E+168 | -104.5 | 2.8E+164 | 2.47E+73 | 8.97112E+19 |
| 106 | 1.1E+170 | -105.5 | -3E+166 | 1.23E+74 | -1.47403E+20 |
| 107 | 1.2E+172 | -106.5 | 3.1E+168 | 6.16E+74 | 2.42227E+20 |
| 108 | 1.3E+174 | -107.5 | -3E+170 | 3.08E+75 | -3.98105E+20 |
| 109 | 1.4E+176 | -108.5 | 3.6E+172 | 1.54E+76 | 6.54378E+20 |
| 110 | 1.6E+178 | -109.5 | -4E+174 | 7.7E+76 | -1.07576E+21 |
| 111 | 1.8E+180 | -110.5 | 4.3E+176 | 3.85E+77 | 1.7687E+21 |
| 112 | 2E+182 | -111.5 | -5E+178 | 1.93E+78 | -2.90835E+21 |
| 113 | 2.2E+184 | -112.5 | 5.3E+180 | 9.63E+78 | 4.78291E+21 |
| 114 | 2.5E+186 | -113.5 | -6E+182 | 4.81E+79 | -7.86663E+21 |
| 115 | 2.9E+188 | -114.5 | 6.7E+184 | 2.41E+80 | 1.294E+22 |
| 116 | 3.4E+190 | -115.5 | -8E+186 | 1.2E+81 | -2.12878E+22 |
| 117 | 4E+192 | -116.5 | 8.9E+188 | 6.02E+81 | 3.50249E+22 |
| 118 | 4.7E+194 | -117.5 | -1E+191 | 3.01E+82 | -5.76327E+22 |
| 119 | 5.6E+196 | -118.5 | 1.2E+193 | 1.5E+83 | 9.48438E+22 |
| 120 | 6.7E+198 | -119.5 | -1E+195 | 7.52E+83 | -1.56097E+23 |
| 121 | 8.1E+200 | -120.5 | 1.7E+197 | 3.76E+84 | 2.56937E+23 |
| 122 | 9.9E+202 | -121.5 | -2E+199 | 1.88E+85 | -4.22963E+23 |
| 123 | 1.2E+205 | -122.5 | 2.5E+201 | 9.4E+85 | 6.96341E+23 |
| 124 | 1.5E+207 | -123.5 | -3E+203 | 4.7E+86 | -1.14653E+24 |
| 125 | 1.9E+209 | -124.5 | 3.8E+205 | 2.35E+87 | 1.88795E+24 |
| 126 | 2.4E+211 | -125.5 | -5E+207 | 1.18E+88 | -3.10913E+24 |
| 127 | 3E+213 | -126.5 | 6E+209 | 5.88E+88 | 5.12067E+24 |
| 128 | 3.9E+215 | -127.5 | -8E+211 | 2.94E+89 | -8.43444E+24 |
| 129 | 5E+217 | -128.5 | 9.6E+213 | 1.47E+90 | 1.38939E+25 |
| 130 | 6.5E+219 | -129.5 | -1E+216 | 7.35E+90 | -2.28894E+25 |
| 131 | 8.5E+221 | -130.5 | 1.6E+218 | 3.67E+91 | 3.77122E+25 |
| 132 | 1.1E+224 | -131.5 | -2E+220 | 1.84E+92 | -6.21393E+25 |
| 133 | 1.5E+226 | -132.5 | 2.7E+222 | 9.18E+92 | 1.02398E+26 |
| 134 | 2E+228 | -133.5 | -4E+224 | 4.59E+93 | -1.68752E+26 |
| 135 | 2.7E+230 | -134.5 | 4.9E+226 | 2.3E+94 | 2.78129E+26 |
| 136 | 3.7E+232 | -135.5 | -7E+228 | 1.15E+95 | -4.58435E+26 |
| 137 | 5E+234 | -136.5 | 8.8E+230 | 5.74E+95 | 7.55693E+26 |
| 138 | 6.9E+236 | -137.5 | -1E+233 | 2.87E+96 | -1.2458E+27 |
| 139 | 9.6E+238 | -138.5 | 1.7E+235 | 1.43E+97 | 2.05392E+27 |
| 140 | 1.3E+241 | -139.5 | -2E+237 | 7.17E+97 | -3.38653E+27 |
| 141 | 1.9E+243 | -140.5 | 3.2E+239 | 3.59E+98 | 5.58417E+27 |
| 142 | 2.7E+245 | -141.5 | -5E+241 | 1.8E+99 | -9.20863E+27 |
| 143 | 3.9E+247 | -142.5 | 6.4E+243 | 9E+99 | 1.51867E+28 |
| 144 | 5.6E+249 | -143.5 | -9E+245 | 4.5E+100 | -2.50476E+28 |
| 145 | 8E+251 | -144.5 | 1.3E+248 | 2.2E+101 | 4.13141E+28 |
| 146 | 1.2E+254 | -145.5 | -2E+250 | 1.1E+102 | -6.81494E+28 |
| 147 | 1.7E+256 | -146.5 | 2.7E+252 | 5.6E+102 | 1.12423E+29 |
| 148 | 2.6E+258 | -147.5 | -4E+254 | 2.8E+103 | -1.85473E+29 |
| 149 | 3.8E+260 | -148.5 | 5.9E+256 | 1.4E+104 | 3.0601E+29 |
| 150 | 5.7E+262 | -149.5 | -9E+258 | 7E+104 | -5.04916E+29 |
| 151 | 8.6E+264 | -150.5 | 1.3E+261 | 3.5E+105 | 8.33168E+29 |
| 152 | 1.3E+267 | -151.5 | -2E+263 | 1.8E+106 | -1.37491E+30 |
| 153 | 2E+269 | -152.5 | 3E+265 | 8.8E+106 | 2.26905E+30 |
| 154 | 3.1E+271 | -153.5 | -5E+267 | 4.4E+107 | -3.74491E+30 |
| 155 | 4.8E+273 | -154.5 | 7E+269 | 2.2E+108 | 6.18112E+30 |
| 156 | 7.5E+275 | -155.5 | -1E+272 | 1.1E+109 | -1.02028E+31 |
| | | | sum | -6.35368E+30 |
울프램 알파로도 계산해봤는데
이렇게 나오네용...
감마함수는 뭔지도 모르겠고...
어디서 부터 잘못된거고 어떻게 풀어야하죠...????ㅜㅜㅜ
