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Those infinite small steps (Posted on 2006-10-03) Difficulty: 2 of 5
Show that 1+1/2+1/3+1/4+...is infinite.

No Solution Yet Submitted by Art M    
Rating: 3.6667 (3 votes)

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re: One proof out of many | Comment 2 of 8 |
(In reply to One proof out of many by JLo)

Yours is the only proof I know of.  This is equivalent but may be clearer.

Note that the series can be broken up by powers of two:

1    +   1/2   +  1/3+1/4  +  1/5+1/6+1/7+1/8   +   1/9+...+1/16   +   ...

Which is greater than

1   +   1/2   +   1/4+1/4   +   1/8+1/8+1/8+1/8  +   1/16+...+1/16   +   ...  =  1 + 1/2 + 1/2 + 1/2 + 1/2 + ...

An expression that is clearly infinite.  If our expression is greater than this it must also be infinite.


  Posted by Jer on 2006-10-03 11:00:20
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