`a)``A=1/(1+3)+1/(1+3+5)+1/(1+3+5+7)+...+1/(1+3+5+7+...+2023)`

`A=1/2^2+1/3^2+1/4^2+...+1/1012^2`

Nhận xét:

`1/3^2<1/2.3`

`1/4^2<1/3.4`

`...`

`1/1012^2<1/1011.1012`

`=>A<1/2^2+1/2.3+1/3.4+...+1/1011.1012`

`=>A<1/4+1/2-1/3+1/3-1/4+...+1/1011-1/1012`

`=>A<1/4+1/2-1/1012`

`=>A<3/4-1/1012<3/4`

Vậy `A<3/4`(đpcm)

`b)``B=1/2-1/4+1/8-1/16+1/32-1/64`

`B=1/2-1/2^2+1/2^3-1/2^4+1/2^5-1/2^6`

`2B=2.(1/2-1/2^2+1/2^3-1/2^4+1/2^5-1/2^6)`

`2B=1-1/2+1/2^2-1/2^3+1/2^4-1/2^5`

`2B+B=(1-1/2+1/2^2-1/2^3+1/2^4-1/2^5)+(1/2-1/2^2+1/2^3-1/2^4+1/2^5-1/2^6)`

`3B=1-1/2^6`

`B=1/3-1/(2^6. 3)<1/3`

Vậy `B<1/3`(đpcm)

`c)``C=1/31+1/32+1/33+...+1/60`

`C=(1/31+1/32+1/33+...+1/45)+(1/46+1/47+1/48+...+1/60)`

Nhận xét:

`1/31+1/32+1/33+...+1/45>1/45+1/45+1/45+...+1/45=1/3`

`1/46+1/46+1/47+...+1/60>1/60+1/60+1/60+...+1/60=1/4`

`=>C>1/3+1/4`

`=>C>4/12+3/12`

`=>C>7/12`

Vậy `C>7/12`(đpcm)