单位分数求和

王守恩

单位分数求和
1，求证。下面所有单位分数的和小于1。
    1/2^3+1/3^3+1/4^3+1/5^3+1/6^3.........
    1/3^3+1/4^3+1/5^3+1/6^3+1/7^3.........
    1/4^3+1/5^3+1/6^3+1/7^3+1/8^3.........  
    1/5^3+1/6^3+1/7^3+1/8^3+1/9^3.........  
    1/6^3+1/7^3+1/8^3+1/9^3+1/10^3.......
    .......................


2，求证。下面所有单位分数的和是无穷大。
    1/3^2+1/4^2+1/5^2+1/6^2+1/7^2.........
    1/4^2+1/5^2+1/6^2+1/7^2+1/8^2.........  
    1/5^2+1/6^2+1/7^2+1/8^2+1/9^2.........  
    1/6^2+1/7^2+1/8^2+1/9^2+1/10^2.......  
    1/7^2+1/8^2+1/9^2+1/10^2+1/11^2.....
    .......................

manthanein

https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constant

hujunhua

第1题, 分3步
1）如果你不想使用定积分的话，可以证明一个裂项不等式`\frac1{(n+1)^3}<\frac1{2n^2}-\frac1{2(n+1)^2}`
2）第 i 行裂项相消得到其和`<\frac1{2i^2}`，并再次裂项`\frac1{2i^2}<\frac1{2(i-1)}-\frac1{2i}`
3）所以各行的结果再相加相消的结果就是小于1.

第2题，仿上先证明一个裂项不等式`\frac1{(n+2)^2}>\frac1{n+2}-\frac1{n+3}`, 然后第 i 行裂项相消得其和`>\frac1{i+2}`, 于是所有行按顺序是一个调和数列，其和发散。

northwolves

\[ \lim_{n \to \infty} \sum_{k=3}^n \frac{1}{k^2} = \frac{\pi^2}{6}-\frac{5}{4}\]

王守恩

单位分数求和
1，求证。下面所有单位分数的和小于1。
    1/2^3+1/3^3+1/4^3+1/5^3+1/6^3.........
    1/3^3+1/4^3+1/5^3+1/6^3+1/7^3.........
    1/4^3+1/5^3+1/6^3+1/7^3+1/8^3.........  
    1/5^3+1/6^3+1/7^3+1/8^3+1/9^3.........  
    1/6^3+1/7^3+1/8^3+1/9^3+1/10^3.......
    .......................


2，求证。下面所有单位分数的和是无穷大。
  
    1/2^2+1/3^2+1/4^2+1/5^2+1/6^2.........
    1/3^2+1/4^2+1/5^2+1/6^2+1/7^2.........
    1/4^2+1/5^2+1/6^2+1/7^2+1/8^2.........  
    1/5^2+1/6^2+1/7^2+1/8^2+1/9^2.........  
    1/6^2+1/7^2+1/8^2+1/9^2+1/10^2.......
    .......................


3，问：当n=?时，下面所有单位分数的和等于1。
    1/2^n+1/3^n+1/4^n+1/5^n+1/6^n.........
    1/3^n+1/4^n+1/5^n+1/6^n+1/7^n.........
    1/4^n+1/5^n+1/6^n+1/7^n+1/8^n.........  
    1/5^n+1/6^n+1/7^n+1/8^n+1/9^n.........  
    1/6^n+1/7^n+1/8^n+1/9^n+1/10^n.......
    .......................

王守恩

\(\D\frac{1}{2}=\frac{\frac{1}{1+2521}+\frac{1}{2+2522}+\frac{1}{3+2523}+\frac{1}{4+2524}+\cdots+\frac{1}{1260+3780}}{1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots+\frac{1}{2519}-\frac{1}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+1681}+\frac{1}{2+1682}+\frac{1}{3+1683}+\frac{1}{4+1684}+\cdots+\frac{1}{1680+3360}}{1+\frac{1}{2}-\frac{2}{3}+\frac{1}{4}+\frac{1}{5}-\frac{2}{6}+\cdots+\frac{1}{2518}+\frac{1}{2519}-\frac{2}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+1261}+\frac{1}{2+1262}+\frac{1}{3+1263}+\frac{1}{4+1264}+\cdots+\frac{1}{1890+3150}}{1+\frac{1}{2}+\frac{1}{3}-\frac{3}{4}+\cdots+\frac{1}{2517}+\frac{1}{2518}+\frac{1}{2519}-\frac{3}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+1009}+\frac{1}{2+1010}+\frac{1}{3+1011}+\frac{1}{4+1012}+\cdots+\frac{1}{2016+3024}}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{4}{5}+\cdots+\frac{1}{2516}+\frac{1}{2517}+\frac{1}{2518}+\frac{1}{2519}-\frac{4}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+841}+\frac{1}{2+842}+\frac{1}{3+843}+\frac{1}{4+844}+\cdots+\frac{1}{2100+2940}}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{5}{6}+\cdots+\frac{1}{2515}+\frac{1}{2516}+\frac{1}{2517}+\frac{1}{2518}+\frac{1}{2519}-\frac{5}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+721}+\frac{1}{2+722}+\frac{1}{3+723}+\frac{1}{4+724}+\cdots+\frac{1}{2160+2880}}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\frac{6}{7}+\cdots+\frac{1}{2514}+\frac{1}{2515}+\frac{1}{2516}+\frac{1}{2517}+\frac{1}{2518}+\frac{1}{2519}-\frac{6}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+631}+\frac{1}{2+632}+\frac{1}{3+633}+\frac{1}{4+634}+\cdots+\frac{1}{2205+2835}}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}-\frac{7}{8}+\cdots+\frac{1}{2013}+\frac{1}{2014}+\frac{1}{2015}+\frac{1}{2016}+\frac{1}{2017}+\frac{1}{2018}+\frac{1}{2019}-\frac{7}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+561}+\frac{1}{2+562}+\frac{1}{3+563}+\frac{1}{4+564}+\cdots+\frac{1}{2240+2800}}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}-\frac{8}{9}+\cdots+\frac{1}{2012}+\frac{1}{2013}+\frac{1}{2014}+\frac{1}{2015}+\frac{1}{2016}+\frac{1}{2017}+\frac{1}{2018}+\frac{1}{2019}-\frac{8}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+501}+\frac{1}{2+502}+\frac{1}{3+503}+\frac{1}{4+504}+\cdots+\frac{1}{2270+2770}}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\frac{9}{10}+\cdots+\frac{1}{2011}+\frac{1}{2012}+\frac{1}{2013}+\frac{1}{2014}+\frac{1}{2015}+\frac{1}{2016}+\frac{1}{2017}+\frac{1}{2018}+\frac{1}{2019}-\frac{9}{2520}}\)

lihpb00

看不懂问题。
    1/3^2+1/4^2+1/5^2+1/6^2+1/7^2.........
这个级数是收敛的