Question on the proof about simple harmonic series..

Question

This is humble proof about harmonic series on my own.

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

= 1 + (1 - 1/2) + {(1 - 1/2) - (1/2 - 1/3)} + {(1 - 1/2) - (1/2 - 1/3) - (1/3 - 1/4)} +...

= 1 + (1/2)n - (1/6)(n-1) - (1/12)(n-2) - (1/20)(n-3)+.....

= 1 + (1/2 - 1/6 - 1/12 - 1/20 -.....)n + (1/6 + 1/6 + 3/20 + 4/30 +....)

and n=>unlimited, therefore harmonic series is divergent.

(add explanation)

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

= 1

 + (1 - 1/2)

 + (1 - 1/2) - (1/2 - 1/3) 

 + (1 - 1/2) - (1/2 - 1/3) - (1/3 - 1/4)

 + (1 - 1/2) - (1/2 - 1/3) - (1/3 - 1/4) - (1/4 - 1/5)..
 .
 .

= 1 + (1/2)n - (1/6)(n-1) - (1/12)(n-2) - (1/20)(n-3)+.....

= 1 + (1/2 - 1/6 - 1/12 - 1/20 -.....)n + (1/6 + 1/6 + 3/20 + 4/30+....)

and n=>unlimited, harmonic series is divergent, too.

but I think it is out of my capability to identify the validity of this expansion

so, give me a correct judgement about my own speculation.

Answer 1

Your proof, as I understand it, is not valid, as it relies on rearranging the series elements.

In fact, for a large class of series you can find a rearrangement such that it converges to an arbitrary element. This is the content of the Riemann series theorem.