1 P1,P2坐标分别是(x1,x1^2)(x2,x2^2) R1=x1^2,R2=x2^2
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由于圆1和圆2相切,根据勾股定理
(P1P2)^2=(R1+R2)^2=(y1-y2)^2+(x1-x2)^2
(x1^2+x2^2)^2=(x1^2-x2^2)^2+(x1-x2)^2
(x1-x2)^2=4x1^2x2^2 x1-x2=2x1x2
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所以(1/x2)-(1/x1)=(x1-x2)/x1x2 =2
同理(1/x3)-(1/x2)=(x2-x3)/x2x3 =2 ...
所以 1/x1 ,1/x2 ,1/x3 ...成等差数列
由于 x1=1 所以x2=1/3 x3=1/5 x4=1/7
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2 S1=πx1^4 S2=πx2^4 ...
约去√π 证明x1^2+x2^2+x3^2+.< 3/2 即可
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x1^2+x2^2+x3^2+.
=1+1/3^2 +1/5^2 +1/7^2 +...
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因为1+1/2^2 +1/3^2 +1/4^2 +1/5^2 +.
=(1+1/3^2 +1/5^2 +1/7^2+...)+(1/4)(1+1/2^2 +1/3^2 +1/4^2 +1/5^2...)
就是 1+1/3^2 +1/5^2 +1/7^2+...=(3/4)(1+1/2^2 +1/3^2 +1/4^2 +1/5^2...)
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又因为 1+(1/2^2 +1/3^2)+(1/4^2 +1/5^2 +1/6^2 +1/7^2)+1/8^2 +...