Question

Chứng minh:

1/2^2 + 1/3^2 + 1/^2+...+1/n^2 <1

Answer 1

Thiếu `n>0`

`1/2^2<1/(1.2)`

`1/3^2<1/(2.3)`

`.........`

`1/n^2<1/((n-1).n)`

`=>VT<1/(1.2)+1/(2.3)+.....+1/((n-1).n)`

`=>VT<1-1/2+1/2-1/3+...+1/(n-1)-1/n=1-1/n<1`

Answer 2

Ta co:\(\dfrac{1}{2^2}< \dfrac{1}{1.2},\dfrac{1}{3^2}< \dfrac{1}{2.3},...,\dfrac{1}{n^2}< \dfrac{1}{\left(n-1\right)n}\)

\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{\left(n-1\right)n}\)

\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{n-1}-\dfrac{1}{n}\)

\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}< 1-\dfrac{1}{n}\)

\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}< 1\left(dpcm\right)\)