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S = 1/3+1/5+1/7+...+1/2013-(1/2+1/4+1/6+...+1/2012)
S = 1/2+1/3+1/4+...+1/2012+1/2013 - 2(1/2+1/4+1/6+...+1/2012)
S = 1/2+1/3+1/4+...+1/2012+1/2013 - (1+1/2+1/3+...+1/1006)
S = 1/1007+1/1008+...+1/2013-1
=> S - P = 1/1007+1/1008+...+1/2013-1-(1/1007+1/1008+...+1/2013)
<=> S - P= -1 <=> (S-P)2013 = -1
Ta có:\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2011}-\frac{1}{2012}\)
=\(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2011}+\frac{1}{2012}-2.\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2012}\right)\)
=\(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2012}-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{1006}\right)\)
=\(\frac{1}{1007}+\frac{1}{1008}+...+\frac{1}{2012}\)
=>\(\left(\frac{A}{B}\right)^{2013}\)=(\(\frac{\frac{1}{1007}+\frac{1}{1008}+...+\frac{1}{2012}}{\frac{1}{1007}+\frac{1}{1008}+...+\frac{1}{2012}}^{ }\))2013=12013=1
\(S=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2011}-\frac{1}{2012}+\frac{1}{2013}\)
\(S=\left(1+\frac{1}{3}+\frac{1}{5}+.....+\frac{1}{2011}+\frac{1}{2013}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{2012}\right)\)
\(S=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+.....+\frac{1}{2011}+\frac{1}{2012}+\frac{1}{2013}\right)-2\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{2012}\right)\)
\(S=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+..+\frac{1}{2011}+\frac{1}{2012}+\frac{1}{2013}-\left(1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{1006}\right)\)
\(S=\frac{1}{1007}+\frac{1}{1008}+.....+\frac{1}{2012}+\frac{1}{2013}=P\)
=>S-P=0
=>(S-P)2016=0
A=1-(1-1/2)+1/3-(1/2-1/4)+..-(1/1006-1/2012)
A=1-1+1/2+1/3-1/2+1/4+...-1/1006+1/2012
A=(1-1)+(1/2-1/2)+...+(1/1006-1/1006)+1/1007+1/1008+..+1/2012
A=B => (A/B)^2013=1
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