■収束するやせざるや(その6)

 一般に,

  1/1−1/2+1/3−1/4+・・・=log2

の項の順序を,正の項をm個,負の項をn個ずつ交互に並べ替えてできる級数の和は

  log2+1/2・logm/n

となる.

[1]m=2,n=1→3/2log2

[2]m=1,n=2→1/2log2

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(証明)並べ替えた数列

{(1/1+1/3+1/5+・・・+1/(2m−1))−(1/2+1/4+1/6+・・・+1/2n)}+{(1/(2m+1)+1/(2m+3)+1/(2m+5)+・・・+1/(4m−1))−(1/(2n+2)+1/(2n+4)+1/(2n+6)+・・・+1/4n)}+・・・

のk次部分和は,

{(1/1−1/2+1/3−1/4+1/5+・・・+1/(2km−1)−1/2km)+(1/2+1/4+1/6+・・・+1/2km)−(1/2+1/4+1/6+・・・+1/2kn}

=log2+o(1/km)+1/2(log(km)+logγ+o(1/km))−1/2(log(kn)+logγ+o(1/kn)

=log2+1/2・log(m/n)+o(1/km)+o(1/kn)

}+{(1/(2m+1)+1/(2m+3)+1/(2m+5)+・・・+1/(4m−1))−(1/(2n+2)+1/(2n+4)+1/(2n+6)+・・・+1/4n)}+・・・

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