欧拉发明了一个调和级数的求和公式，同时也把调和级数引向了它的反面，下面用实算证明：
∑1/n=1+0.5+0.3333+0.25+0.2 → 0,
∑lnn+c=0.577+1.27+1.67+1.96 → ∞,
看看，一个驱近于零的数值，变成了一个驱向于无穷大的式子。一般来说，驱近于零的数值是收敛的式子；驱向于无穷大的式子，就是发散的式子。这样就面对了一个问题，调和级数到底是收敛的，还是发散的。
由于受一些纯数学的影响，比如：兔子永远追不上乌龟的，无限级数，永远会的数值的产生，等等，有的人就会认为调和级数应该是发散的，也用一种类比的方法证明了调和级数是发散的。
这样证明调和级数发散的式子就越来越多。我想在历史上肯定也有证明，调和级数是收敛的式子，由于在众多的证明调和级数发散的式子中，被打压下去了。留给我们的就只能是，调和级数发散的式子。现在看来，有二十多种。
其实这二十多种证明调和级数发散的式子，都只能算是左证，因为它们是采用类比的方法和近似的方法来证明的，也就是说用别的一些数学手段来证明它。而不是用自身的规律，来证明自身。
看起来很吓人的，二十多种方法。其实用实算法和数学规律，分分钟就把它们都证伪了。

调和级数真要辩证看待，对于数学理论来说，发散。对于实际问题来说，收敛

欧拉公式：(k=1,n)∑1/k=lnn+c
用你臆想的欧拉公式来攻击欧拉，惨无人道！

不懂装懂的两种人，与三江方士有一比。

美国有一名句：无理攻击你的人越多，越证明你的成就伟大！

我们说话都是有根据的：

看清楚：∑1/n≈lnn+c。

欧拉给出来的常数是一个固定的常数，但实际的常数是一个变化的：
.....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
这是数学规律，你们是没有办法否定的。

用快算法得出来的收敛结果：
.......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
.....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
..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
通过快算法就能看出来，常数是一个变化的，由1一步步的驱近于零，这就是它的数学规律，这是没有办法否定的。当变化常数为零的时候就收敛了！

你这贴不是五年前发过吗

什么是左证，就是借用别的数学手段来证明：
∑1/n＞∑1/2,
∑1/n＞ln(n+1),
∑1/n＞∫1/xdx,
..................
利用这样的不等式，左边的大于右边的，然后在证明右边的是发散的，然后推理出来左边也是发散的。事实上这样的左证法是违反逻辑的，因为它们并不是同性质的。
∑1/n=1+0.5+0.3333+0.25+0.2 → 0,
∑1/2=1/2+1/2+1/2+1/2 →1/2,
S=ln(n+1)=0.619+1.09+1.39 →∞,
S=lnn+c=0.577+1.27+1.67+1.96 → ∞,
这样也能看出来，他们的性质是相反的，相反的性质是不能进行类比的，不然就违反了逻辑规律。
∑1/2也只能通过玩小把戏，才能让它大于∑1/n，那事实上就是一种骗局；
ln(n+1）这个也只是在开始一段时间内小于∑1/n，当超过四百兆兆后，就开始在于∑1/n；
左证只是理论上的证明，这种理论上的证明还不能违反逻辑规律，违反了逻辑规律必然会是错误的。
再说理论证明，并不是全过程的展示，所以，它也没有什么说服力。
后面还会一个个的来揭穿它们的骗局。