Question

Chứng minh rằng:

\(S=\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+...+\frac{1}{100^2}<\frac{1}{2}\)

Giải nhanh nhanh giùm mình nha, 25/3 mình kiểm tra 45' rồi

Answer 1

\(S=\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+.......+\frac{1}{100^2}<\frac{1}{2}\)

\(S=\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+........+\frac{1}{100^2}\)<\(\frac{1}{0.2}+\frac{1}{2.4}+\frac{1}{4.6}+.......+\frac{1}{98.100}\)

\(S=\frac{1}{2}-\frac{1}{100}=\frac{49}{100}<\frac{50}{100}=\frac{49}{100}<\frac{1}{2}\)

Vậy \(\frac{49}{100}<\frac{1}{2}\)

Answer 2

Ta có 1/2²<1/2*3

         1/4²<1/3*4

         . . .

         1/100²<1/99*100

=> S<1/2*3+1/3*4+...+1/99*100

=> S<1/2-1/3+1/3-1/4+...+1/99-1/100

=>S<1/2-1/100

=>S<49/100

Mà 49/100<1/2

=>S<1/2

Answer 3

               S = 1/2^2 + 1/4^2 + 1/6^2 + ... + 1/100^2 

suy ra:   4*S =    1   + 1/2^2 + 1/3^2 + ... + 1/50^2

có: 1/2^2 = 1/2*2 < 1/1*2

     1/3^2 = 1/3*3 < 1/2*3 

      1/50^2 = 1/50*50 <1/49*50

1+ 1/2^2 + 1/3^2 + ... + 1/50^2 < 1 + 1/1*2 + 1/2*3 + ... +1/49*50

                 4*S< 1 + 1 - 1/2 + 1/2 - 1/3 + ... + 1/49 - 1/50

                  4*S < 2 - 1/50 = 99/50

                S < 99/50 : 4 = 99/50 * 1/4 = 99/200 < 100/200 = 1/2

                                            vậy S < 1/2 (đpcm)