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| 作者 | 標題: 超難的數學 |
didi  IP Address: [ 59.104.223.72 ] |
 發表於: 2011/1/20 下午 08:36:55
9*23*37*51...*2011 = ? 知道是等差級數，但是是求積.....?????? 懇求賜教! |
| 正名 IP Address: [ 118.171.24.149 ] |
 回覆於: 2011/1/20 下午 10:57:10
9*23*37*51...*2011 = ? 知道各項是等差"數列"，但是是求積.....?????? "級數"有和的意義 |
| QQ IP Address: [ 60.251.55.7 ] |
回覆於: 2011/1/21 上午 01:12:58
一個414位數的整數 http://www.wolframalpha.com/input/?i=product+%2814i%2B9%29%2Ci%3D0+to+143 |
| didi IP Address: [ 211.74.237.110 ] |
回覆於: 2011/1/21 上午 11:48:01
非常謝謝! 真是太強了!!也就是說它的十位數是" 2 "嗎? 但是可以再說些有關於式子開頭的數學知識嗎?? 那個符號代表什麼? 是不是就像是"Σ"是連加，而它是連乘? 唸法為何? 謝謝賜教囉!!! |
| n IP Address: [ 124.218.78.148 ] |
回覆於: 2011/1/22 上午 07:02:58
連乘符號為Π(如同Σ一樣，它須要有適當的參數) 唸作pi，為希臘大寫字母第16個(小寫即為π) 在word上方的﹕插入、特殊符號、希臘符號，可找得到 |
| Abel IP Address: [ 211.76.49.69 ] |
回覆於: 2011/1/22 上午 07:36:01
這是用數學軟體 Mathematica 計算出來的 |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 08:29:22
設y=(大寫pi)(k=1to144)(14k-5)=9*23*...*2011 則ln(y)=sigma(k=1to144)ln(14k-5) =ln(9)+ln(23)+...+ln(2011) |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 08:36:23
積分ln(14x-5)dx =(1/14)積分ln(14x-5)d(14x-5) =(1/14)(14x-5)ln(14x-5) -(1/14)積分(14x-5)dln(14x-5) =(1/14)(14x-5)ln(14x-5) -積分(14x-5)/(14x-5)dx =(1/14)(14x-5)ln(14x-5)-積分1dx =(1/14)(14x-5)ln(14x-5)-x+c |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 08:39:20
以下用泰勒定理模仿黎曼積分(以提升準確度) |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 08:47:52
設f(x)=(1/14)(14x-5)ln(14x-5)-x f’(x)=ln(14x-5)=ln(x-(5/14))+ln(14) f’’(x)=(x-(5/14))^(-1) f’’’(x)=-(x-(5/14))^(-2) f’’’’(x)=2(x-(5/14))^(-3) 由於只求近似值,只用泰勒定理的首5項 f(x)=f(a)+f’(a)(x-a)+(1/2)f’’(a)(x-a)^2 +(1/3!)f’’’(a)(x-a)^3+(1/4!)f’’’’(a)(x-a)^4 |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 09:01:33
當x=n+0.5,a=n時,則x-a=1/2 (1/14)(14n+2)ln(14n+2)-(n+0.5) =(1/14)(14n-5)ln(14-5)-n+(1/2)ln(14n-5) +(1/2)(n-(5/14))^(-1)(1/2)^2 -(1/3!)(n-(5/14))^(-2)(1/2)^3 +(1/4!)(n-(5/14))^(-3)(1/2)^4 ____ (1/2)ln(14n-5)=(1/14)(14n+2)ln(14n+2) -(1/14)(14n-5)ln(14n-5)-0.5 -(1/8)(n-(5/14))^(-1) +(1/48)(n-(5/14))^(-2) -(1/192)(n-(5/14))^(-3) |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 09:11:55
更正: +(1/4!)2(n-(5/14)^(-3)(1/2)^4 ___ |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 09:22:06
當x=n-0.5,a=n時,則x-a=-1/2 (1/14)(14n-12)ln(14n-12)-(n-0.5) =(1/14)(14n-5)ln(14n-5)-n-(1/2)ln(14n-5) +(1/2)(n-(5/14))^(-1)(1/2)^2 +(1/3!)(n-(5/14))^(-2)(1/2)^3 +(1/4!)2(n-(5/14)^(-3)(1/2)^4 ___ (1/2)ln(14n-5)=(1/14)(14n-5)ln(14n-5) -(1/14)(14n-12)ln(14n-12)-0.5 +(1/8)(n-(5/14))^(-1) +(1/48)(n-(5/14))^(-2) +(1/192)(n-(5/14))^(-3) |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 09:27:32
兩式相加,得 ln(14n-5)=(1/14)(14n+2)ln(14n+2) -(1/14)(14n-12)ln(14n-12)-1 +(1/24)(n-(5/14)^(-2) |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 09:37:50
例子(1): ln(14×7-5)用(1/14)(14×7+2)ln(14×7+2) -(1/14)(14×7-12)ln(14×7-12)-1 +(1/24)(7-(5/14))^(-2)表示 ___ ln(14×7-5)=ln(93)=4.5325994932 (1/14)(14×7+2)ln(14×7+2) -(1/14)(14×7-12)ln(14×7-12)-1 +(1/24)(7-(5/14))^(-2) =(100/14)ln(100)-(86/14)ln(86) +(1/24)(7-(5/14))^(-2) =4.532597884 ___ 誤差=4.5325994932-4.532597884=1.6092×10^(-6) |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 10:03:36
ln(14n-5)的首6項直接加,不要用上式,以減少誤差 即ln(9)+ln(23)+ln(37)+ln(51)+ln(65) +ln(79)=21.419297461 ln(14×7-5)用(1/14)(14×7+2)ln(14×7+2) -(1/14)(14×7-12)ln(14×7-12)-1 +(1/24)(7-(5/14))^(-2)表示 ln(14×8-5)用(1/14)(14×8+2)ln(14×8+2) -(1/14)(14×8-12)ln(14×8-12)-1 +(1/24)(8-(5/14))^(-2)表示 ... ln(14n-5)用(1/14)(14n+2)ln(14n+2) -(1/14)(14n-12)ln(14n-12)-1 +(1/24)(n-(5/14))^(-2)表示 |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 10:13:30
另外發現:sigma x=1至n時 黎曼積分修正下限0.5上限n+0.5,使誤差減少 (不在這裡討論會太長) |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 10:28:29
例子(2): 當p>q時,(p,q為正整數) 求g(p,q)=(q-(5/14))^(-2) +(q+1-(5/14))^(-2) +...+(p-(5/14)^(-2)=? 設f(x)=(x-(5/14))^(-2) 以下用修正上下限的黎曼積分 g(p,q)=積分(下限q-0.5上限p+0.5)f(x)dx =(下限q-0.5上限p+0.5)-(x-(5/14)^(-1) =(q-(6/7))^(-1)-(p+(1/7))^(-1) |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 10:42:27
例子(3): 當整數n>6時, 求ln(9)+ln(23)+...+ln(14n-5)=? ln(9)+ln(23)+...+ln(14n-5) =21.419297461+(1/14)(14n+2)ln(14n+2) -(1/14)(14×7-12)ln(14×7-12) -(n-6)+(1/24)g(n,7) =21.419297461+(1/14)(14n+2)ln(14n+2) -(86/14)ln(86)-n+6 +(1/24)(7-(6/7))^(-1) -(1/24)(n+(1/7))^(-1) =0.0636613012+(1/14)(14n+2)ln(14n+2) -n-(1/24)(n+(1/7))^(-1) 其中 21.419297461-(86/14)ln(86)+6 +(1/24)(7-(6/7))^(-1)=0.0636613013 |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 10:54:04
例子(4): 求ln(y)=ln(9)+ln(23)+...+ln(2011)=? 當n=144時,則14n-5=2011 ln(y)=0.0636613012 +(1/14)(14×144+2)ln(14×144+2) -144-(1/24)(144+(1/7))^(-1) =952.9706523392 y=e^952.9706523392 log(y)=952.9706523392log(e) =413.8698957267 y=(10^0.8698957267)(10^413) =7.4113227545×10^413 |
| 小昭 IP Address: [ 14.0.158.137 ] |
回覆於: 2024/11/15 上午 11:07:51
用電腦excel計算: ln(y)=ln(9)+ln(23)+...+ln(2011) =952.9706419 y=e^952.9706419 log(y)=952.9706419log(e) =413.869891193 y=(10^0.869891193)(10^413) =7.4112453864×10^413 ln(y)的誤差=952.9706523392 -952.9706419=1.04392×10^(-5) y的百分比誤差=100%×7.4113227545×10^413 /7.4112453864×10^413-100% =0.00104393% |
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