Question

Cho A = \(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+.....+\(\frac{1}{2014^2}\)+\(\frac{1}{2015^2}\)+\(\frac{1}{2016^2}\)

CHỨNG MINH RẰNG A KHÔNG PHẢI LÀ SỐ TỰ NHIÊN.

Answer 1

\(A=\frac{1}{2^2}+\frac{1}{3^2}+.........+\frac{1}{2016^2}\)

\(\frac{1}{2^2}<\frac{1}{1\cdot2}\)

\(\frac{1}{3^2}<\frac{1}{2\cdot3}\)

...........

\(\frac{1}{2016^2}<\frac{1}{2015\cdot2016}\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+........+\frac{1}{2016^2}<\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+.....+\frac{1}{2015\cdot2016}\)

\(\Rightarrow A<\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.....+\frac{1}{2015}-\frac{1}{2016}\)

\(\Rightarrow A<\frac{1}{1}-\frac{1}{2016}\)

\(\Rightarrow A=\frac{2015}{2016}\)

\(\Rightarrow A<1\)    (1)

\(\frac{1}{2^2}>0\)

\(\frac{1}{3^2}>0\)

........

\(\frac{1}{2016^2}>0\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+......+\frac{1}{2016^2}>0+0+.......+0\)

\(\Rightarrow A>0\)       (2)

Từ (1) và (2):

\(\Rightarrow\)0<A<1

\(\Rightarrow\)A không là số tự nhiên