so sánh: \(A=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...........+\frac{1}{3^{2015}}\)
với \(B=\frac{1}{2}\)
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\(0,1\left(3\right)=\frac{1,\left(3\right)}{10}=\frac{1+0,\left(3\right)}{10}=\frac{1+\frac{1}{3}}{10}=\frac{\frac{4}{3}}{10}=\frac{4}{30}\)
a/P=1-1/2+1/3-1/4+1/5-1/6+...+1/199-1/200
=(1+1/3+1/5+1/7+...+1/199)-(1/2+1/4+1/6+...+1/200)
=(1+1/2+1/3+1/4+1/5+1/6+...+1/99+1/200)-2(1/2+1/4+1/6+...+1/200)
=(1+1/2+1/3+1/4+1/5+1/6+...+1/99+1/200)-(1+1/2+1/3+...+1/100)
=1/101+1/102+1/103+...+1/200
a/ P=1-1/2+1/3-1/4+....+1/199-1/200
= 1+1/2+1/3+1/4+1/5+...+1/200 - 2.(1/2+1/4+...+1/200)
= 1+1/2+1/3+1/4+1/5+...+1/200 - 1-1/2-1/3-...-1/100
=1/101+1/102+...+1/200
b/ k-k/2+ k/3- k/4+...+k/199-k/200
=k+k/2+k/2+...+k/199+k/200 -2(k/2+k/4+k/6+...+k/200)
=k+k/2+k/2+...+k/199+k/200-k-k/2-k/3-...-k/100
=k/101+k/102+...+k.200
\(3A=1+\frac{1}{3}+.......+\frac{1}{3^{2014}}\)
\(3A-A=\left(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2014}}\right)-\left(\frac{1}{3}+\frac{1}{3^2}+....+\frac{1}{3^{2015}}\right)\)
\(\Rightarrow2A=1-\frac{1}{2015}\)
\(\Rightarrow A=\left(1-\frac{1}{3^{2015}}\right):2=\frac{1}{2}-\frac{1}{3^{2015}.2}