柯西收敛准则 求柯西收敛准则的具体意义和实例啊。写的具体点。实例中的思想。

2025年01月25日 14:11
有2个网友回答
网友(1):

定理叙述:数列{xn}有极限的充要条件是:对任意给定的ε>0,有一正整数N,当m,n>N时,有|xn-xm|<ε成立   
将柯西收敛原理推广到函数极限中则有:函数f(x)在无穷远处有极限的充要条件是:对任意给定的ε>0,有Z属于实数,当x,y>Z时,有|f(x)-f(y)|<ε成立   
此外柯西收敛原理还可推广到广义积分是否收敛,数项级数是否收敛的判别中,有较大的适用范围。

证明举例:
证明:xn=1-1/2+1/3-1/4+......+ [(-1)^(n+1)]/n 有极限   
证:
对于任意的m,n属于正整数,m>n
|xn-xm|=| [(-1)^(n+2)]/(n+1)+......+[(-1)^(m+1)]/m |   
当m-n为奇数时 |xn-xm|=| [(-1)^(n+2)]/(n+1)+......+[(-1)^(m+1)]/m |
<1/n(n+1)+1/(n+1)(n+2)+......+1/(m-1)m   
=(1/n-1/m)→0   
由柯西收敛原理得{xn}收敛   
当m-n为偶数时 |xn-xm|=| [(-1)^(n+2)]/(n+1)+......+[(-1)^(m+1)]/m |   
<1/n(n+1)+1/(n+1)(n+2)+......+1/(m-2)(m-1)-1/m   
=(1/n-1/(m-1)-1/m)→0   
由柯西收敛原理得{xn}收敛   
综上{xn}收敛,即{xn}存在极限

网友(2):

这都不会???丢人!百度一下全都有了

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