cho 1/1^2 + 1/2^2 +1/3^2+...+1/10^2.hay so sanh M voi 1/1/3
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\(M=\frac{1}{1.2}+\frac{2}{1.2.3}+.....+\frac{9}{1.2.3.....10}\)
\(M=\frac{2-1}{1.2}+\frac{3-1}{1.2.3}+....+\frac{10-1}{1.2......10}\)
\(M=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{6}+....+\frac{10}{1.2.....10}-\frac{1}{1.2.....10}\)
\(M=1-\frac{1}{1.2.3......10}\)
\(M=1-\frac{1}{3628800}\)
Vì \(1=1\)mà \(\frac{1}{3628800}< 1\)nên \(1-\frac{1}{3628800}< 1\)
Vậy \(M< 1\)
\(M=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{10^2}\)
\(>1+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{10.11}\)
\(=1+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{10}-\frac{1}{11}\)
\(=1+\frac{1}{2}-\frac{1}{11}\)
\(>1+\frac{1}{2}-\frac{1}{6}=\frac{4}{3}\)
\(A=\left(\frac{1}{2^2}-1\right).\left(\frac{1}{3^2}-1\right)......\left(\frac{1}{10^2}-1\right)=\left(-\frac{3}{4}\right).\left(-\frac{8}{9}\right)......\left(-\frac{99}{100}\right)\)
\(A=\frac{\left(-3\right).\left(-8\right).....\left(-99\right)}{4.9........100}=\frac{\left(-1\right).3.\left(-2\right).4....\left(-9\right).11}{2.2.3.3.....10.10}=\frac{\left[\left(-1.-2.-3....-9\right).\left(3.4...11\right)\right]}{\left(2.3.....10\right).\left(2.3...10\right)}\)
\(A=\frac{\left(-1\right).11}{10.2}=\frac{-11}{20}< \frac{-10}{20}=\frac{-1}{2}\)
Suy ra \(A< -\frac{1}{2}\)
\(M=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....+\frac{1}{2^{2016}}\)
\(\Rightarrow\)\(2M=2+1+\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^{2015}}\)
\(\Rightarrow\)\(2M-M=\left(2+1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2015}}\right)-\left(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2016}}\right)\)
\(\Rightarrow\)\(M=2-\frac{1}{2^{2016}}< 2\)
Vậy M < 2
\(3a=3+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2018}}\)
\(2a=3a-a=3-\frac{1}{3}-\frac{1}{3^{2019}}< 3\Rightarrow a< \frac{3}{2}\)
\(M=\frac{1}{1^2}+\frac{1}{2^2}+..+\frac{1}{10^2}\)
\(M=1+\frac{1}{2^2}+...+\frac{1}{10^2}\)
\(\Rightarrow M>1+\frac{1}{2.3}+...+\frac{1}{10.11}\)
\(\Rightarrow M>1+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{10}-\frac{1}{11}\)
\(\Rightarrow M>1+\frac{1}{2}-\frac{1}{11}\)
\(\Rightarrow M>1+\frac{9}{11}=1\frac{9}{22}=1\frac{27}{66}>1\frac{22}{66}=1\frac{1}{3}\)
\(\Rightarrow M>1\frac{1}{3}\)