1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+.....1/n等于多少?
3个回答
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这是调和级数,没有通项公式,有近似公式
1+1/2+1/3+……+1/n=lnn
ln是自然对数,
当n 趋于无穷时,
1+1/2+1/3+……+1/n=lnn+R
R为欧拉常数,约为0.5772.
推理查看百科上有,不知道你能不能看懂
1665年牛顿在他的著名著作《流数法》中推导出第一个幂级数:
ln(1+x) = x - x2/2 + x3/3 - ...
Euler(欧拉)在1734年,利用Newton的成果,首先获得了调和级数有限多项和的值。结果是:
1+1/2+1/3+1/4+...+1/n= ln(n+1)+r(r为常量)
他的证明是这样的:
根据Newton的幂级数有:
ln(1+1/x) = 1/x - 1/2x^2 + 1/3x^3 - ...
于是:
1/x = ln((x+1)/x) + 1/2x^2 - 1/3x^3 + ...
代入x=1,2,...,n,就给出:
1/1 = ln(2) + 1/2 - 1/3 + 1/4 -1/5 + ...
1/2 = ln(3/2) + 1/2*4 - 1/3*8 + 1/4*16 - ...
......
1/n = ln((n+1)/n) + 1/2n^2 - 1/3n^3 + ...
相加,就得到:
1+1/2+1/3+1/4+...1/n = ln(n+1) + 1/2*(1+1/4+1/9+...+1/n^2) - 1/3*(1+1/8+1/27+...+1/n^3) + ......
后面那一串和都是收敛的,我们可以定义
1+1/2+1/3+1/4+...1/n = ln(n+1) + r
Euler近似地计算了r的值,约为0.577218。这个数字就是后来称作的欧拉常数。
1+1/2+1/3+……+1/n=lnn
ln是自然对数,
当n 趋于无穷时,
1+1/2+1/3+……+1/n=lnn+R
R为欧拉常数,约为0.5772.
推理查看百科上有,不知道你能不能看懂
1665年牛顿在他的著名著作《流数法》中推导出第一个幂级数:
ln(1+x) = x - x2/2 + x3/3 - ...
Euler(欧拉)在1734年,利用Newton的成果,首先获得了调和级数有限多项和的值。结果是:
1+1/2+1/3+1/4+...+1/n= ln(n+1)+r(r为常量)
他的证明是这样的:
根据Newton的幂级数有:
ln(1+1/x) = 1/x - 1/2x^2 + 1/3x^3 - ...
于是:
1/x = ln((x+1)/x) + 1/2x^2 - 1/3x^3 + ...
代入x=1,2,...,n,就给出:
1/1 = ln(2) + 1/2 - 1/3 + 1/4 -1/5 + ...
1/2 = ln(3/2) + 1/2*4 - 1/3*8 + 1/4*16 - ...
......
1/n = ln((n+1)/n) + 1/2n^2 - 1/3n^3 + ...
相加,就得到:
1+1/2+1/3+1/4+...1/n = ln(n+1) + 1/2*(1+1/4+1/9+...+1/n^2) - 1/3*(1+1/8+1/27+...+1/n^3) + ......
后面那一串和都是收敛的,我们可以定义
1+1/2+1/3+1/4+...1/n = ln(n+1) + r
Euler近似地计算了r的值,约为0.577218。这个数字就是后来称作的欧拉常数。
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hich the result had not the least practical consequence.Its interest lay in its intricacy and its beauty in its solution. But like purebeauty it led nowhere. His future was blank. He was fifty-five, and full ofenergy, and to himself his mind seemed as alert as ever, his experience of menand affairs was wide: all that remained to him was to settle down in a countrytown in England and play bridge with elderly ladies and golf with retiredcolonels. He had met, when on leave, old chiefs of his, and had observed withwhat difficulty they adapted themselves to the change in their circumstances.They had looked forward to the freedom that would be theirs when they retiredand pictured the charming uses t
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