求一有理數

1/10+1/100+2/1000+3/10000+5/100000+8/1000000+13/10000000+21/100000000+
34/1000000000 + ......
Sol
p=1/10
A=p+p^2+2p^3+3p^4+5p^5+8p^6+13p^7+21p^8+34p^9+……
Ap= p^2+ p^3+2p^4+3p^5+5p^6+8p^7+13p^8+21p^9+……
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A(1－p)=p+ p^3+ p^4+2p^5+3p^6+5p^7+8p^8+13p^9+……
=p+p^2(p+ p^2+2p^3+3p^4+5p^5+8p^6+13p^7+……)
=p+p^2A
So
A－Ap=p+p^2A
A(1－p－p^2)=p
A=p/(1－p－p^2)
=100p/(100－100p－100p^2)
=10/(100－10－1)
=10/89