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\(A=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2019.2020}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2019}-\frac{1}{2020}\)
\(=1-\frac{1}{2020}< 1\)
Vậy \(A< 1\left(đpcm\right)\)
\(B=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}< \frac{1}{4}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)
\(\Leftrightarrow B< \frac{1}{4}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)
\(\Leftrightarrow B< \frac{1}{4}+\frac{1}{2}-\frac{1}{50}\)
\(\Leftrightarrow B< \frac{1}{4}+\frac{1}{2}\)
\(\Leftrightarrow B< \frac{3}{4}\left(đpcm\right)\)
Gọi biểu thức trên là A.
Chứng minh A > 50
\(A=1+\frac{1}{2}+\left(\frac{1}{2^1+1}+\frac{1}{2^2}\right)+\left(\frac{1}{2^2+1}+\frac{1}{6}+...+\frac{1}{2^3}\right)+...+\left(\frac{1}{^{2^{100-2}+1}}+...+\frac{1}{2^{100-1}}\right)\\ \)
\(A>1+\frac{1}{2}+\frac{1}{2^2}.2+\frac{1}{2^3}.2^2+...+\frac{1}{2^{100-1}}2^{100-2}\)
\(A>\left(\frac{1}{2}+\frac{1}{2}\right)+\frac{1}{2}+\frac{1}{2}+...+\frac{1}{2}\)
\(< =>A>\frac{100}{2}=50\)
Chứng minh A<100
\(A=1+\left(\frac{1}{2}+\frac{1}{3}\right)+\left(\frac{1}{2^2}+\frac{1}{5}+...+\frac{1}{7}\right)+....+\left(\frac{1}{2^{100-2}}+\frac{1}{2^{100-2}+1}+...+\frac{1}{2^{100-1}-1}\right)\)-\(\frac{1}{2^{100-1}}\)
\(A< 1+\frac{1}{2}.2+\frac{1}{2^2}.2^2+...+\frac{1}{2^{100-2}}.2^{100-2}+\frac{1}{2^{100-1}}\)
\(A< 1+1+1+...+1+\frac{1}{2^{100-1}}\)
\(A< 1.99+\frac{1}{2^{100-1}}< 99+1=100\)
ta có : 1+1/2+1/3+....+1/2^100-1
= 1/2x2 +1/3x2 +1/4x2 +...+ 1/2^100 x2
= 2x(1/2+1/3+1/4+...+1/2^100)
=.................... làm đến đây mk tịt
Ta có: \(55+5\)1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 +.....+ 1/50^2 = 1/1^2 + 1/2^2 + (1/3^2 + 1/4^2 +....+ 1/50^2 )
< 1 + 1/4 + (1/2*3 + 1/3*4 +...+1/49*50) = 1 + 1/4 + (1/2 - 1/3 + 1/3 - 1/4+...+1/49 - 1/50 )
= 1,73 = 173/100 (dpcm)
\(VP=\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{99}{100}\)
\(VP=\frac{2-1}{2}+\frac{3-1}{3}+\frac{4-1}{4}+...+\frac{100-1}{100}\)
\(VP=\frac{2}{2}-\frac{1}{2}+\frac{3}{3}-\frac{1}{3}+\frac{4}{4}-\frac{1}{4}+...+\frac{100}{100}-\frac{1}{100}\)
\(VP=1-\frac{1}{2}+1-\frac{1}{3}+1-\frac{1}{4}+...+1-\frac{1}{100}\)
\(VP=100-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{100}\right)=VT\) ( đpcm )
Mk nghĩ \(VT=100-\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{100}\right)\) bn xem lại đề có nhầm ko
Chúc bạn học tốt ~
a)\(\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...+\frac{99}{100!}\)
=\(\frac{2}{2!}-\frac{1}{2!}+\frac{3}{3!}-\frac{1}{3!}+\frac{4}{4!}-\frac{1}{4!}+...+\frac{100}{100!}-\frac{1}{100!}\)
=\(1-\frac{1}{2!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{3!}-\frac{1}{4!}+...+\frac{1}{99!}-\frac{1}{100!}\)
=\(1-\frac{1}{100!}< 1\)
\(\Rightarrow\)\(\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...+\frac{99}{100!}< 1\)
b)\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}\)
=\(\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+\frac{3.4}{4!}-\frac{1}{4!}+...+\frac{99.100}{100!}-\frac{1}{100!}\)
=\(\left(\frac{1.2}{2!}+\frac{2.3}{3!}+\frac{3.4}{4!}+...+\frac{99.100}{100!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)\)=\(1+1-\frac{1}{99}-\frac{1}{100}\)
=\(2-\frac{1}{99}-\frac{1}{100}< 2\)
\(\Rightarrow\)\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}< 2\)