Σ1/FnFn+2=1/2+1/3+1/10+1/24+・・・

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Σ1/FnFn+2=Σ(1/Fn-1/Fn+2)/(Fn+2-Fn)=Σ(1/Fn-1/Fn+2)/(Fn+1)＝Σ(1/FnFn+1-1/Fn+1Fn+2)

=(1/F1F2-1/F2F3)+(1/F2F3-1/F3F4)+・・・=1/F1F2=1

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これはΣ1/n!(n+2)=1/2の証明と同様の方法である。

Σ1/n!{1/(n+1)-1/(n+2)}=Σ1/(n+2)!を考える

Σ1/n!{1/(n+1)}=Σ1/(n+1)!

Σ1/n!(n+2)=Σ1/(n+1)!-Σ1/(n+2)!={1/2!+1/3!+1/4!+・・・}-{1/3!+1/4!+1/5!+・・・}

=1/2

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