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Analysis with series 1/2^2+1/3^2
Posted:
Aug 8, 2012 10:23 AM
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Hello teacher~
{(1/2^2) + (1/3^2) + (1/4^2) + ...}
+ {(1/2^3) + (1/3^3) + (1/4^3) + ...}
+ {(1/2^4) + (1/3^4) + (1/4^4) + ...}
+ ...
----------------------------------------------
I have a solution.(ambiguous)
Namely,
{(1/2^2) + (1/3^2) + (1/4^2) + ...}
+ {(1/2^3) + (1/3^3) + (1/4^3) + ...}
+ {(1/2^4) + (1/3^4) + (1/4^4) + ...}
+ ...
=
(1/2^2 + 1/2^3 + 1/2^4 + ...)
+ (1/3^2 + 1/3^3 + 1/3^4 + ...)
+ (1/4^2 + 1/4^3 + 1/4^4 + ...)
(this associative law ? Really possible?)
=
(1/2^2)/{1-(1/2)}
+ (1/3^2)/{1-(1/3)}
+ (1/4^2)/{1-(1/4)}
+...
= (1/2) + (1/3).(1/2) + (1/4).(1/3) + ...
= (1/2) + {(1/2)-(1/3)} + {(1/3)-(1/4)} + ...
= 1
---------------------------------------------
Hm... how do you think about it ?
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