■ある無限級数(その80)

[1]1+1/2+1/4+1/8+1/16+・・・=2

[2]1-1/3+1/5-1/7+1/9+・・・=π/4

[3]1+(i/2)+(i/2)^2+(i/2)^3+・・・

=1/{|1-(i/2)}=4/5+2i/5

[4]1+1/2+1/3+1/4+1/5+・・・=∞

   1+1/2+1/3+1/4+1/5+・・・+1/n~logn

[5]1/2+1/3+1/5+1/7+1/11+1/13+・・・=∞

   1/2+1/3+1/5+1/7+1/11+1/13+・・・+1/p~loglogp

[6]1+1/2^2+1/3^3+1/4^2+1/5^2+・・・=π^2/6

   1+1/2^4+1/3^4+1/4^4+1/5^4+・・・=π^4/90

   1+1/2^3+1/3^3+1/4^3+1/5^3+・・・=???

   1+1/2^p+1/3^p+1/4^p+1/5^p+・・・<p/(p-1)

[7]1-1/2+1/4-1/8+1/16+・・・=2/3

[8]1-1/2+1/3-1/4+1/5+・・・=log2

[9]1+1+1/2!+1/3!+1/4!+1/5!+・・・=e

   1-1+1/2!-1/3!+1/4!-1/5!+・・・=1/e

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