怎样学算法？ - 知乎<strong class="NumberBoard-itemValue" title="1被浏览<strong class="NumberBoard-itemValue" title=",213,568分享邀请回答2.1K171 条评论分享收藏感谢收起gzc/CLRSgithub.com3. 斯坦福Tim Roughgarden教授的算法专项课程。教授人长得帅，声音好听，尤其是课讲得超棒！下面链接是专项课程第一门课 Divide and Conquer, Sorting and Searching, and Randomized Algorithms。这门课是收费的，但是点开专项课程中单独一门课的主页。可以找到申请助学金的选项，按要求填好申请，15天后就会100%通过申请，然后就可以访问课程的全部资源了。4. 一本机械工业出版社影印的书《算法概论》，因为三位作者分别叫Dasgupta，Papadimitriou，Vazirani，人称这本书为DPV。它在不长的篇幅里（大概只有《算法导论》的四分之一）涵盖了基本的算法知识，同时兼顾了严谨性，是本不错的入门书。最后，快过年啦，给我看过的算法书来张全家福，嘿嘿40178 条评论分享收藏感谢收起知乎 - 发现更大的世界Inconspicuous手机验证码登录忘记密码？登录没有帐号？注册下载知乎 App没有更多推荐了，
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神奇的HyperLogLog算法【转载 #涉及到数学原理】
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数学规律和实算法证明∑1/n是收敛的收藏
经过高级计算机和电脑的校对，修改了计算中的错误。下面就是电脑校对后的数据，充分的证明了lnn&#65310;∑1/n，∑1/n是收敛的，实算证明是没有办法否定的，下面就是一步步的实算证明：.....N.........................∑............................lnn..............................标准的变化常数....1......................1.00000.......................0......................................1.00000.....2.....................1.50000...................0.67932................................0.80285.....3.....................1.83333...................1.09861................................0.73472.....4.....................2.83333...................1.38629................................0.69704.....5.....................2.28333....................1.60944...............................0.67389.....6.....................2.45000....................1.79176................................0.65824.....7.....................2.59286.....................1.94591...............................0.64695.....8.....................2.71786.....................2.07944...............................0.63842.....9.....................2.82897.....................2.19722...............................0.63275....10....................2.92897.....................2.30259...............................0.62638.....20...................3.59774.....................2.99573...............................0.60201.....30...................3.99499.....................3.40120...............................0.59379.....40...................4.27854......................3.68888..............................0.58966.....50...................4.49921......................3.91202..............................0.58719.....60...................4.67987......................4.09435..............................0.58719.....70...................4.83284......................4.24850..............................0.58434.....80...................4.96548......................4.38203..............................0.58345.....90...................5.08257......................4.49981..............................0.58276....100..................5.18738......................4.60517...............................0.58221.....200.................5.87803......................5.29831...............................0.57972.....300.................6.28266......................5.70378...............................0.57888.....400.................6.56993......................5.99146...............................0.57847.....500.................6.79282......................6.21460...............................0.57822.....600.................6.97498......................6.39693...............................0.57805.....700.................7.12901......................6.55108...............................0.57793.....800.................7.26245......................6.68461...............................0.57784.....900.................7.38017......................6.80240...............................0.57777....1000................7.48547......................6.90776...............................0.57771....2000................8.17837......................7.60092...............................0.57745....3000...............8.58375.......................8.00637...............................0.57738只要电脑上有多功能计算器的，上面就有算∑值的，小学水平的人，都能看得懂，都能证明。
.......N.........................∑...........................lnn...........................变化常数....4000..................8.87075.................8.29405......................0.57670....5000..................9.09342.................8.51719......................0.57623....6000..................9.27515.................8.69952......................0.57563....7000..................9.42882.................8.85367......................0.57515....8000..................9.56196.................8.98719.......................0.57487....9000..................9.67958.................9.10498.......................0.57460....一万...................9.78467.................9.21034.......................0.57433....二万..................10.47584................9.90349.......................0.57235....三万..................10.87719...............10.30895......................0.56824....四万..................11.16213...............10.59663......................0.56550....五万..................11.38340...............10.81978......................0.56362....六万..................11.56438...............11.00210......................0.56228...七万...................11.71740...............11.15625......................0.56115...八万...................11.85016...............11.28978......................0.56036...九万...................11.96733...............11.40757......................0.55976...十万...................12.07219...............11.51293......................0.55926..二十万................12.76336...............12.20678......................0.55658..三十万................13.16471...............12.61154......................0.55317..四十万................13.44965...............12.89922......................0.55043..五十万................13.67092...............13.12236......................0.54856..六十万................13.85190...............13.30469......................0.54721..七十万................14.00502...............13.45884......................0.54618..八十万................14.13778...............13.59237......................0.54541..九十万.................14.25495..............13.71015......................0.54480...百万...................14.35981...............13.81551......................0.54430..二百万................15.05098................14.50866.....................0.54232..三百万................15.45233................14.91412.....................0.53821..四百万................15.73727................15.20181.....................0.53546我想用数学编程最高也只能算到这里，你如果懂的话，就可以用来校对。相差最多也是在近似值范围内。
..N.............................∑..........................lnn......................变化常数..五百万................15.95854...............15.42499.................0.53355..六百万................16.13952...............15.60727.................0.53225..七百万................16.29264...............15.76142.................0.53122..八百万................16.42540...............15.89495.................0.53045..九百万................16.54257...............16.01274.................0.52983..一千万................16.64743...............16.11800.................0.52933..二千万................17.33860...............16.81124.................0.52324..三千万................17.73995...............17.21671.................0.52324..四千万................18.02489...............17.50439.................0.52050..五千万................18.24616...............17.72753.................0.51966..六千万................18.42714................17.90986................0.51728..七千万................18.58026...............18.06401.................0.51625..八千万................18.71302...............18.19754.................0.51548..九千万................18.83016...............18.31532.................0.51487..一亿...................18.93505................18.42068................0.51437..二亿...................19.62622................19.11383................0.51239..三亿...................20.02757................19.51929................0.50828..四亿...................20.31251................19.80698................0.50553..五亿...................20.53378................20.03012................0.50366..六亿...................20.71476................20.21244................0.50232..七亿...................20.86788................20.36659................0.50129..八亿...................21.00064................20.50012................0.50052..九亿...................21.11781................20.61791................0.49990..十亿...................21.22267................20.72327................0.49940..二十亿...............21.91384................21.41641.................0.49743..三十亿...............22.31419................21.82188.................0.49231这都是采用快算法算的：∑1/n=.................101+1/00+1/2................万位至万亿位采用以上的方法。..
你怎么时算证明了？既然你声称自己证明了，那∑1/n收敛在多少？
呵呵 好聪明啊 楼主 去发表吧 但愿别让保安轰出来
什么高级计算机？什么电脑？用什么算？
这些数据能说明什么?
（1）快算法的准确度：十位数的算法：∑1/n=................10（1/61+1/70）1/2，∑1/N（70）=60+10（1/61+1/70）1/2。N................快算值...................∑1/n值70.............4.83328.................4.8328680.............4.96618.................4.9655190.............5.08348.................5.08257100...........5.18757.................5.18743百位数的算法：N(200)-N(600)得算法：∑1/n=...............100（1/101+1/167+1/200）1/3，∑1/N（200）=100+100（1/101+1/167+1/200）1/3。N...................快算值...............∑1/n值200...............5.88390.............5.87810300...............6.28590.............6.28298400...............6.57313.............6.57025500...............6.79440.............6.79315600...............6.97528.............6.97498N（700）-N（1000）后采用：∑1/n=...............100（1/601+1/650+1/700）1/3，N..................快算值................∑1/n值700..............7.12964..............7.12901800..............7.26332..............7.26245900..............7.38129..............7.380171000............7.48671..............7.48547千位算法：∑1/n=...............1+1/0）1/3，N..................快算值...............∑1/n值2000.............8.19671..............8.178373000.............8.60261..............8.58375.............从表中可以看出来，快算法有点偏高，偏会拉高N得收敛数。万位算法：∑1/n=101+1/0）1/3光凭这个快算法，虽然可以算到很高位数，但太浪费时间。快算中还有快算才是解决问题的关键。
.....N...........................∑...........................lnn.............................变化化常数..四十亿................22.59913................22.10956........................0.48957..五十亿................22.82040................22.33275........................0.48765..六十亿................23.00138................22.51503........................0.48635..七十亿................23.15450................22.66960........................0.48490..八十亿................23.28726................22.80271........................0.48455..九十亿................23.40443................22.92049........................0.48394....百亿.................23.50929................23.02585........................0.48344..二百亿...............24.20046.................23.71900........................0.48146..三百亿...............24.60181.................24.12445........................0.47736..四百亿...............24.88675.................24.41215........................0.47460..五百亿...............25.10802.................24.63629........................0.47273..六百亿...............25.28900................24.81761.........................0.47139..七百亿...............25.44212................25.10559.........................0.47036..八百亿...............25.57488................25.10559.........................0.46959..九百亿...............25.69205................25.22308.........................0.46897...千亿.................25.79691.................25.32844.........................0.46847..二千亿..............25.48808.................26.02158..........................0.46650..三千亿..............26.88943.................26.42705..........................0.46238..四千亿..............27.17437.................26.71473..........................0.45964..五千亿..............27.39564.................26.93787..........................0.45777..六千亿..............27.57662.................27.12020..........................0.45642..七千亿..............27.72974.................27.27435..........................0.45539..八千亿..............27.86250.................27.40788..........................0.45462...九千亿.............27.97967.................27.52366..........................0.45401...万亿.................28.06453................27.63102..........................0.45351...二万亿.............28.77570.................28.32417..........................0.45153..三万亿..............29.17705.................28.72963..........................0.44742
.....N................................∑.................................lnn..............................变化常数..四万亿.....................29.46199.....................29.01732.........................0.44467..五万亿.....................29.68326.....................29.24045.........................0.44281..六万亿.....................29.86424.....................29.42278.........................0.44146..七万亿.....................30.01736.....................29.57693.........................0.44043..八万亿.....................30.15012.....................29.71046.........................0.43966..九万亿.....................30.26729.....................29.82825..........................0.43904...十万亿....................30.37215.....................29.93361.........................0.43854..二十万亿..................31.06057.....................30.62675..........................0.43382..三十万亿..................31.46086.....................31.03222..........................0.42864..四十万亿..................31.72524.....................31.31990..........................0.42534..五十万亿..................31.66917.....................31.54304..........................0.42313..六十万亿..................32.14692.....................31.72536..........................0.42156..七十万亿..................32.29990.....................31.87954..........................0.42036..八十万亿..................32.43253.....................32.01305..........................0.41948..九十万亿..................32.54959.....................32.13083..........................0.41874...百万亿.....................32.65437.....................32.23619...........................0.41828..二百万亿....................33.34279......................32.92933..........................0.41346..三百万亿....................33.74308......................33.33480..........................0.40828..四百万亿....................34.02746......................33.62248..........................0.40498..五百万亿....................34.24839......................33.84563..........................0.40276..六百万亿....................34.42914......................34.02795..........................0.40119..七百万亿....................34.58212......................34.18210..........................0.40002..八百万亿....................34.71475......................34.21533..........................0.39942..九百万亿....................34.83181......................34.43342..........................0.39839..千万亿.......................34.93659......................34.53878...........................0.39781..二千万亿....................35.62501.....................35.23192............................0.39309..三千万亿....................36.02530.....................35.63731............................0.38799..四千万亿....................36.30968.....................35.92507............................0.38461.五千万亿.....................36.53061.....................36.14821............................0.38240.六千万亿.....................36.71136.....................36.33054............................0.38081.七千万亿.....................36.86434.....................36.48469............................0.37965
（76）相对论，大爆炸理论，错就错在它们没有从本质看问题，从整体的科学规律看问题，从整体的宇宙运行规律看问题。而是从表面的，不能确定的一些现象，脱离规律，脱离本质的去猜想，这种猜想带有很强的主观性，而不是科学性。所以，它自然会走向错误，走向荒谬！巧合不等于规律，巧合也不等于真理，巧合有它的偶然性，表面的相同性，它并非是本质的共同表现。要透过现象看事物的本质，才能真正的明白一切。相对论，大爆炸理论，就是相信这些表面的巧合性，就是相信这些表现的共同性。才会被巧合所欺骗，才会被表面的现象所麻醉，把巧合当真理，把表面现象当规律。结果就犯下了，一个个的错误！运动的本质因素是什么，物质的本质规律是什么，宇宙稳定的因素是什么，这些问题都没有搞清楚，我们怎么可能找到正确的思路，找到正确的方向！比如说，光速能让时钟停止，时钟能等于时间吗？光速能不能让时钟停止，这还是个问题。人坐在光速的飞船里，人就永远年轻吗？人是一个生命体，他适不适应光速还是一个问题。飞机的速度并不快，飞行员必须要身体素质最好的人。盲目的联想，就会犯主观的错误；盲目的猜想，就会犯脱离客观的错误。
我一直想知道你的快算怎么来的……拍脑袋随手写的吗……证明呢？
宋卫芳是个大傻逼！
........N...............................∑.............................lnn............................变化常数..八千万亿.....................36.99697................36.61822......................0.37875..九千万亿.....................37.11403................36.73600......................0.37803....亿亿..........................37.21881................36.84362......................0.37519...二亿亿........................37.90723...............37.55451.......................0.37272...三亿亿........................38.30752...............37.93997.......................0.36755...四亿亿........................38.59190...............38.22766.......................0.36324...五亿亿........................38.81283...............38.45080.......................0.36203...六亿亿........................38.99358...............38.63312.......................0.36043...七亿亿........................39.14656...............38.78716.......................0.35940...八亿亿........................39.27932...............39.92080.......................0.35852...九亿亿........................39.39649...............39.03859........................0.35790...十亿亿........................39.50135...............39.14395........................0.35740..二十亿亿.....................40.18977...............39.83709........................0.35268..三十亿亿.....................40.59006...............40.24256........................0.34750..四十亿亿.....................40.87444...............40.53024........................0.34420..五十亿亿.....................41.09537...............40.75339........................0.37198..六十亿亿.....................41.27612...............40.73206........................0.34106..七十亿亿.....................41.42910...............41.08986........................0.33924..八十亿亿.....................41.56173...............41.22339........................0.33834..九十亿亿.....................41.67890...............41.34117........................0.33773...百亿亿.......................41.78376................41.44653.......................0.33723.二百亿亿......................42.47218...............42.13968........................0.33250.三百亿亿......................42.87247...............42.54514........................0.32733.四百亿亿......................43.15741...............42.83283........................0.32458.五百亿亿......................43.37834..............43.05597.........................0.32237.六百亿亿......................43.55909...............43.52829........................0.32080.七百亿亿......................43.71207...............43.29244........................0.31963.八百亿亿......................43.84470...............43.52597........................0.31873.九百亿亿......................43.96176...............43.64376........................0.31790....千亿亿......................44.06652...............43.74912........................0.31740.二千亿亿.....................44.75494...............44.44226........................0.31268.三千亿亿.....................45.15523...............44.84773........................0.30750
不可能收敛的
原式=lnx+y+o（1）趋向于无完
无穷，打错字了
百度说调和级数是发散的，这玩意我信，你那才算几项？人家算到了10的40多次方，10的后面40个零。“前100万项相加约为14.357；前10亿项相加约为21；前一万亿项相加约为28” 欧拉已经给出了公式，还给出了公式的证明过程。而且这个级数1360年别人就证明了它是发散的，构造了一个级数，这个级数的每一项都比调和级数的对应项小，所以总和比调和级数小，但这个构造的数列是发散的，调和级数就肯定是发散的。
这两个级数的大小一比较就出来了，后面的级数是发散的，它相当于无穷多个1/2相加。1 +
1/2+ 1/3 +1/4 + 1/5+ 1/6+1/7+1/8 +...1/2+1/2+(1/4+1/4)+(1/8+ 1/8+1/8+1/8)+...
没看明白，根据你列的数据，显然能给人该级数是发散的感觉。
lnn是趋近于无穷大的吧？它比调和级数大显然不能证明调和级数是收敛的。
几百年来，关于∑1/n由于没有直接的求和公式，无法知道他最终的结果，也就无法定认它是收敛，还是发散。所以就只能采用类比的方法来证明，看它是收敛，还是发散。由于这些类比存在问题，就导致了一些错误的结论。导致这些错误的根源就是违反了数学规律。从书上和网上来看，大约有六、七种方法，下面将用数学规律来一一证明它们是错误的。（1）第一个错误方法是：1+1/2+1/3+1/4+1/5...........&#/2+1/2+1/2+1/2+1/2.............这一看就知道是错误，下面用实算来证：但是有人在这里玩了两个小把戏，下面来看看这两个小把戏：第一小把戏：他是把第二1/2分成两位，把第三个1/2分成四位，这样有同位上，前者就比后者大了，这纯粹属于骗子的小把戏，我们用同样的方法就能否定它。S1=∑1/N=1+(1/8+1/8+1/8+1/8)+(1/12+1/12+1/12+1/12)+(1/16+1/16+1/16+1/16)..........S1=∑1/N=1+1/2+(1/9+1/9)+(1/12+1/12+1/12+1/12).............S1&#65308;S2,因为S1本质就比S2，分解后还是比S2小。
只能说明你根本就不懂数学！
你说得很对，欧拉算得大数，跟我的是一样的，反过来证明我快算法是正确的！
.....N..................................∑..................................lnn................................变化常数..二亿亿兆..................74.42497........................74.37587..........................0.04910..三亿亿兆..................74.82520........................74.78134..........................0.04386..四亿亿兆..................75.10958........................75.06902..........................0.04056..五亿亿兆..................75.33051........................75.29216..........................0.03835..六亿亿兆..................75.51126........................75.47448..........................0.03678..七亿亿兆..................75.66424........................75.62833..........................0.03591..八亿亿兆..................75.79687........................75.76217..........................0.03470..九亿亿兆..................75.91393........................75.98531..........................0.03399..十亿亿兆..................76.01871........................75.98531..........................0.03340..二十兆兆..................76.70713........................76.67846..........................0.02567..三十兆兆..................77.10737........................77.08392..........................0.02345..四十兆兆..................77.39175........................77.37160..........................0.02016..五十兆兆..................77.61268........................77.59475..........................0.01793.六十兆兆...................77.79343........................77.77707..........................0.01536.七十兆兆..................77.94641.........................77.93122..........................0.01519.八十兆兆..................78.07904.........................78.06496..........................0.01408.九十兆兆..................78.19610.........................78.18253..........................0.01357.百亿亿兆..................78.30088.........................78.28787..........................0.01299.二百兆兆..................78.98830.........................78.98104..........................0.00726.三百兆兆..................79.38859.........................79.28651..........................0.00208.四百兆兆..................79.67297.........................79.67419..........................0.00000
数学规律推断：（1）∑1/n→0，lnn→∞,虽然开始∑1/n比lnn，它们的增长规律也有点相似，但是∑1/n→0，lnn→∞，∑1/n的增值量会越来越小，而lnn却是恒量增长的，所以，最终lnn会大于∑1/n；（2）常数C是随着N值的变化而变化的，从1→0，欧拉的不变常数是错误的；（3）当C为零的时候，就进入了收敛范围，因为C为负数时，增长值会越变越小，违反了数学规律，所以就必须开始收敛。光是用数学规律来推断有的人是不服的，上面用实算也证明了数学规律的推断是完全正确的。实算证明：（1）演变出来了变化常数C由1→0的整个过程；（2）证明了，N为四百兆兆的时候，lnn&#65310;∑1/n，往后走差值会越来越大，因为∑1/n会进入无限小，收敛。∑1/n=...................∞（1/∞+1/∞+1/∞+1/∞）1/4=∞/∞=1。在铁的事实面前，你们还敢说假话吗！
也可以用三江方士的一种求和方法来左证明：以上就是三江的原方法，算起来比较的麻烦，我将它进行过多次的完善，现在比较的理想了，可能还要在不断的研究中进一步完善。
我现在发现五角败类们真是太不要脸了，都不知道你们在我们眼里就是垃圾和废物，我写东西给你们看，你们也真敢把自己当个人！你们看得懂吗，你们看了又有什么用呢！你们只是一群垃圾和疯子，把你们当病人看，理都不理你们，还天天跑来，看我的贴子，你们真敢自己当人的话，就永远别进我跟我有关的贴子。如果进来，就是你们自己骂自己是狗！也是你们自己承认你们自己跪在我的脚 下，学狗叫了！你们害怕我发贴子，才是真实的。你们越是害怕，我们就越是要发，不把你们气疯，气死，算我没有本事！我发贴子，是给真正懂得数学的学者们看的。五角败类们请自己要脸，永远别进我的贴子！
看看那些搞全盘西化的败类们，连常识都不懂，也有脸胡言乱语，我们不得不用科学知识来羞辱他们。他们都不知道C为什么不能为零。下面就一一的来分析：S=∑1/n→a,如果a不在变了的化，就说明它的Sn不会在增长了，自然就收敛了；怎么证明它是收敛的呢，Un=S(n)-S(n-1),当Un=0时，收敛，因为它的Sn不再增长。∑1/n&#65308;=&#65310;f(n),Sn=∫1/f(n),∫1/f(n)=lnn+c.∑1/n=lnn+c.n=400*10^32时，Sn=79.67297,lnn=79.67419,C=∑1/n-lnn=79.19=-0.00122∑1/n=lnn+c.设：N=^29,c=-0.00123,lnn=79.67419,∑1/n=lnn+c=79.23=79.67296Un=S(n)-S(n-1)=^29-^29=79.97=-0,00001,收敛。当C为负数时，Sn变小了。
你之前的S1和S2两个式子中的n的意义都不同，典型的伪证，具体的我都不想说什么了。你是欺负现在的计算机的处理数据速度还没那么大是吧，等百年后的计算机的计算速度够了后，那时希望大家谈起民科的时候还能想起你们。
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