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đặt A=1/2^2+1/3^2+1/4^2+...+1/100^2
B=1/2.3+1/3.4+...+1/99.100
=1/1.2+1/2.3+1/3.4+...+1/99.100
=1-1/2+1/2-1/3+...+1/99-1/100
=1-1/100<1 (1)
Mà 1<2(2)
A =1/1+1/2.2+1/3.3+...+1/100.100<1-1/2+1/2-1/3+...+1/99-1/100 (3)
từ (1),(2),(3) =>A<2
\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}<\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{99.100}\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}<1-\frac{1}{100}<1\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}<1\)
C=1/2*2+1/4*4+1/6*6+...+1/100*100.
C<1/4+1/2*4+1/4*6+1/6*8+...+1/98*100.
C<1/4+1/2*(2/2*4+2/4*6+2/6*8+...+2/98*100).
C<1/4+1/2*(1/2-1/4+1/4-1/6+1/6-1/8+...+1/98-1/100).
C<1/4+1/2*(1/2-1/100).
C<1/4+1/2*49/100.
C<1/4+49/200.
C<1/4+50/200=1/2.
Vậy C<1/2.
ta có \(\frac{1}{2\cdot2}+\frac{1}{4\cdot4}+\frac{1}{6\cdot6}+.........+\frac{1}{100\cdot100}\)
\(< \frac{1}{4}+\frac{1}{2x4}+\frac{1}{4\cdot6}+\frac{1}{6\cdot8}+........+\frac{1}{98\cdot100}\)
\(\frac{1}{4}+\frac{1}{2}\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+......+\frac{1}{98\cdot100}\right)\)
=\(\frac{1}{4}+\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{100}\right)=\frac{1}{4}+\frac{1}{2}\cdot\frac{49}{100}=\frac{1}{4}+\frac{49}{200}\)
tự làm nốt
mình chỉ gợi ý thôi, vì viết cái này mỏi tay lắm thông cảm nha
Ở phần ''a'' bạn hãy đổi ra thành:2=2;4=2;.....sau dó bạn CM \(\frac{1}{2^2}<\frac{1}{1.2}.....\) rồi hãy suy ra nhỏ hơn \(\frac{1}{3}\)
còn phần ''b'' bạn hãy tách ra nha
dễ mà mình làm hoài hà bạn nhân A cho \(\frac{1}{3}\)rồi sau đó cộng A và \(\frac{1}{3}\times A\) lại tiếp theo tự tính
c) \(M=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{99}{100}< \frac{1}{2}.\frac{4}{4}.\frac{6}{6}...\frac{100}{100}=\frac{1}{2}\)
Dat A=1/3-2/32+3/33-4/34+...+99/399-100/3100
3A=1-2/3+3/32-4/33+...+99/398-100/399
3A+A=1-1/3+1/32-1/33+...+1/398-1/399-100/3100=4A
4A.3=3-1+1/3-1/32+...+1/397-1/398-100/399=12A
4A+12A=3-100/399-1/399-100/3100
16A=3-300/3100-3/3100-100/3100=3-403/3100<3
A<3/16
Chung to...
Ta có :
\(\frac{1}{3^2}< \frac{1}{2\times3};\frac{1}{4^2}< \frac{1}{3\times4};\frac{1}{5^2}< \frac{1}{4\times5};\frac{1}{6^2}< \frac{1}{5\times6};...;\frac{1}{100^2}< \frac{1}{99\times100}\)
\(\Rightarrow\) \(\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}< \frac{1}{2\times3}+\frac{1}{3\times4}+\frac{1}{4\times5}+\frac{1}{5\times6}+...+\frac{1}{99\times100}\)
\(\Rightarrow\) \(\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}< \frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{100}\)
\(\Rightarrow\) \(\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}< \frac{1}{2}-\frac{1}{100}\)
\(\Rightarrow\) \(\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}< \frac{49}{100}< \frac{50}{100}=\frac{1}{2}\)
\(\Rightarrow\) \(\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}< \frac{1}{2}\)
\(B=\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}< \frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{99.100}\)
\(=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{99}-\frac{1}{100}=\frac{1}{4}-\frac{1}{100}< \frac{1}{4}\)
\(B=\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}>\frac{1}{5.6}+\frac{1}{6.7}+...+\frac{1}{100.101}\)
\(=\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{100}-\frac{1}{101}=\frac{1}{5}-\frac{1}{101}=\frac{1}{6}+\frac{1}{6.5}-\frac{1}{101}>\frac{1}{6}\)
=> \(\frac{1}{6}< B< \frac{1}{4}\)
Ta có : \(\frac{1}{2^2}<\frac{1}{1\cdot2}\)
\(\frac{1}{3^2}<\frac{1}{2\cdot3}\)
...
\(\frac{1}{100^2}<\frac{1}{99\cdot100}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+.......+\frac{1}{100^2}<\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+.......+\frac{1}{99\cdot100}\)
Ta có : \(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+.......+\frac{1}{99\cdot100}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.......+\frac{1}{99}-\frac{1}{100}\)
\(=1-\frac{1}{100}<1\)
\(\Rightarrow\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+.......+\frac{1}{99\cdot100}<1\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+.......+\frac{1}{100^2}<1\Rightarrow1+\frac{1}{2^2}+\frac{1}{3^2}+.......+\frac{1}{100^2}<2\)
Ta có
1 + 1/2^2 + 1/3^2+.....+1/100^2 = 1,634939
=)) 1,634939 < 2