调和级数与欧拉常数
形如：![\[\sum_{i=1}^{n}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\]](/user_files/SPECODER/epics/d744f05e5e7bd84791b946a013611a4731f16a35.png "\[\sum_{i=1}^{n}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\]")
的级数称作调和级数。
数百年前，数学家们就对其极限长生了兴趣：
即：![\[\lim_{n\rightarrow+\infty}\sum_{i=1}^{n}\frac{1}{i}\]](/user_files/SPECODER/epics/aa3eeddf1fda7fa922a007b04bc0cf8f758de51a.png "\[\lim_{n\rightarrow+\infty}\sum_{i=1}^{n}\frac{1}{i}\]")
是否存在。（n ,i 均为自然数）
下面给出最简单也是最显而易见的证明：
当![\[n\rightarrow+\infty\]](/user_files/SPECODER/epics/555d765eed872184a42b0671a8efb106cba030fd.png "\[n\rightarrow+\infty\]")
时
显然![\[\lim_{n\rightarrow+\infty}\sum_{i=1}^{n}1\]](/user_files/SPECODER/epics/8f3af676a10c40a62ba3a18a3c4673616eaed699.png "\[\lim_{n\rightarrow+\infty}\sum_{i=1}^{n}1\]")
是不存在的。
之后，数学家们又发现了其他的一些性质，其中最著名的就是欧拉常数了：
![\[\lim_{n\rightarrow+\infty}\Big[\sum_{i=1}^{n-1}\frac{1}{i}-\ln(n)\Big]\approx0.57721\]](/user_files/SPECODER/epics/4c222e686ee074540aa2b15f92c7f858df7cc9d9.png "\[\lim_{n\rightarrow+\infty}\Big[\sum_{i=1}^{n-1}\frac{1}{i}-\ln(n)\Big]\approx0.57721\]")
左式最早由欧拉在1740年，证明极限存在，且估算出了小数点后5位。
下面让我们来证明，左式极限存在，用的是最简单的方法，即证：左式单调有界。
1）单调性：
令：![\[a_n=\sum_{i=1}^{n-1}\frac{1}{i}-\ln(n)=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n-1}-\ln(n)\]](/user_files/SPECODER/epics/d5a9b38d6aeb36f533ea88b1204ee227b0fd78ef.png "\[a_n=\sum_{i=1}^{n-1}\frac{1}{i}-\ln(n)=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n-1}-\ln(n)\]")
则：![\[a_{n+1}=\sum_{i=1}^{n}\frac{1}{i}-\ln(n+1)=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln(n+1)\]](/user_files/SPECODER/epics/0f4ddef1ae4e632f6d7f1f13bbd4ca63bc596f86.png "\[a_{n+1}=\sum_{i=1}^{n}\frac{1}{i}-\ln(n+1)=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln(n+1)\]")
![\[a_{n+1}-a_{n}=\frac{1}{n}-\ln(\frac{n+1}{n})=\frac{1}{n}\times1-\int_{n}^{n+1}\frac{1}{x} d(x)\]](/user_files/SPECODER/epics/b0bc23663fb390f40640b6343fe365b95a072418.png "\[a_{n+1}-a_{n}=\frac{1}{n}-\ln(\frac{n+1}{n})=\frac{1}{n}\times1-\int_{n}^{n+1}\frac{1}{x} d(x)\]")
为了判别上式的符号，我们利用数形结合：
我们可以看到：
![\[\int_{n}^{n+1}\frac{1}{x} d(x)\]](/user_files/SPECODER/epics/1e78e50c8c422fb029262970e55203853364bebf.png "\[\int_{n}^{n+1}\frac{1}{x} d(x)\]")
的面积就是 蓝色的面积+灰色的面积；
![\[\frac{1}{n}\times1\]](/user_files/SPECODER/epics/389c56577275d8cc69b551fd85bd54bd41a34ecb.png "\[\frac{1}{n}\times1\]")
的面积就是 蓝色的面积+灰色的面积+红色的面积；
![\[\frac{1}{n+1}\times1\]](/user_files/SPECODER/epics/92fab39df678b7f0ee5589b8b3df05fe5095a4e9.png "\[\frac{1}{n+1}\times1\]")
的面积就是 蓝色的面积；
显然：![\[\frac{1}{n+1}\,< \int_{n}^{n+1}\frac{1}{x}d(x)\,<\,\frac{1}{n}\]](/user_files/SPECODER/epics/c940e90e800d60e53b1029055da86701e53f2781.png "\[\frac{1}{n+1}\,< \int_{n}^{n+1}\frac{1}{x}d(x)\,<\,\frac{1}{n}\]")
,
即: ![\[\frac{1}{n+1}\,< \ln(\frac{n+1}{n})\,<\,\frac{1}{n}\]](/user_files/SPECODER/epics/5a99797e299c40ad5dc8c7d81fc0d649bb979cc0.png "\[\frac{1}{n+1}\,< \ln(\frac{n+1}{n})\,<\,\frac{1}{n}\]")
所以：![\[a_{n+1}-a_{n}\,>\,0\]](/user_files/SPECODER/epics/b1f78363f5a2392c5fe9018dc4e2a3dcd5972299.png "\[a_{n+1}-a_{n}\,>\,0\]")
结论： ![\[a_{n}\]](/user_files/SPECODER/epics/751a2805def1ede1a11f940cb125573669b340af.png "\[a_{n}\]")
为一个递增序列。
2）有界性：(利用不等式)
结论：为一个有上界的序列。
综合，1）和 2）可知：![\[\lim_{n\rightarrow+\infty}\Big[\sum_{i=1}^{n-1}\frac{1}{i}-\ln(n)\Big]\]](/user_files/SPECODER/epics/994498a6d32dec3e273eb6bc386928730469ba85.png "\[\lim_{n\rightarrow+\infty}\Big[\sum_{i=1}^{n-1}\frac{1}{i}-\ln(n)\Big]\]")
存在，且小于1。
数学界称其为欧拉常数，其值约为0.577
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