Question

[Hoàng Thị Ý Nhi]

A = 1/1²+1/2²+1/3²+...+1/50²

^{Chứng minh A < 2}

Answer 1

[Issac Newton]

Chuyển đổi \(A=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+.....=\frac{1}{49.50}\)

\(A=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{4}+\frac{1}{5}-.......+\frac{1}{49}-\frac{1}{50}\)

\(A=\frac{1}{1}-\left(-\frac{1}{2}+\frac{1}{2}\right)+\left(-\frac{1}{3}+\frac{1}{3}\right)+\left(-\frac{1}{4}+\frac{1}{4}\right)+.......+\left(-\frac{1}{49}+\frac{1}{49}\right)-\frac{1}{50}\)

\(A=\frac{1}{1}-0+0+0+0+.......+0+0-\frac{1}{50}\)

\(A=\frac{1}{1}-\frac{1}{50}=\frac{49}{50}\)

Có \(\frac{49}{50}<2\) nên \(A<2\)

Answer 2

[hoàng tử bóng tối]

                                           GIẢI

A=1/1²  +1/2²  + 1/3²  + ....+ 1/50²

A<1+1/1.2+1/2.3+....+1/49.50

A<1+1-1/2+1/2-1/3+...+1/49-1/50

A<2-1/50<2

vậy A<2