一个收敛无穷级数 paradox

来源: 于 2014-08-13 08:21:15 [博客] [旧帖] [给我悄悄话] 本文已被阅读：次


Note:  All the series on this page are convergent!!!


The series  

1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ......

is a convergent alternating series. Let S be the sum.

Then S = (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + ...    is positive.  .......... (1)



On the other hand,

S = 1 - 1/2 -  1/4 + 1/3 - 1/6 - 1/8 + 1/5 - 1/10 - 1/12 + 1/7 - ......

= (1 - 1/2)  - 1/4 + (1/3 - 1/6) - 1/8 + (1/5 - 1/10) - 1/12 + ...

= 1/2 - 1/4 + 1/6 - 1/8 + 1/10 - 1/12 + .....

= (1/2) [1 - 1/2 + 1/3 - 1/4 + 1/5 -  1/6 + ...... ]

= S/2 .

Hence, S = 0. But this contradicts (1).