Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
\(A=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+....+\dfrac{1}{18.19.20}=\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+...+\dfrac{1}{18.19}-\dfrac{1}{19.20}\right)\\ =\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{19.20}\right)\\ =\dfrac{1}{4}-\dfrac{1}{2.19.20}< \dfrac{1}{4}\)
Cái B TT nhé
\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+....+\dfrac{1}{n^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{\left(n-1\right)n}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{n-1}-\dfrac{1}{n}\\ =1-\dfrac{1}{n}< 1\)
D TT
E mk thấy nó ss ớ
\(S=\left(1-\dfrac{1}{4}\right)+\left(1-\dfrac{1}{9}\right)+\left(1-\dfrac{1}{16}\right)+...+\left(1-\dfrac{1}{n^2}\right)\\ S=\left(1+1+...+1\right)-\left(\dfrac{1}{4}+\dfrac{1}{9}+...+\dfrac{1}{n^2}\right)\\ S=n-1-\left(\dfrac{1}{4}+\dfrac{1}{9}+...+\dfrac{1}{n^2}\right)< n-1\)
Lại có \(\dfrac{1}{4}+\dfrac{1}{9}+..+\dfrac{1}{n^2}=\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}\)
\(\Rightarrow\dfrac{1}{4}+\dfrac{1}{9}+...+\dfrac{1}{n^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{n\left(n-1\right)}< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{n-1}-\dfrac{1}{n}=1-\dfrac{1}{n}< 1\)
\(\Rightarrow S>n-1-1=n-2\\ \Rightarrow n-2< S< n-1\\ \Rightarrow S\notin N\)
\(A=\dfrac{1}{\sqrt{1}}+\dfrac{1}{\sqrt{2}}+...+\dfrac{1}{\sqrt{n}}>\sqrt{n}\left(1\right)\)
Với \(n=2\), BĐT \(\left(1\right)\) trở thành \(\dfrac{1}{\sqrt{1}}+\dfrac{1}{\sqrt{2}}>\sqrt{2}\) (đúng)
Giả sử \(\left(1\right)\) đúng với \(n=k\), nghĩa là \(\dfrac{1}{\sqrt{1}}+\dfrac{1}{\sqrt{2}}+...+\dfrac{1}{\sqrt{k}}>\sqrt{k}\left(2\right)\)
Ta chứng minh \(\left(1\right)\) đúng với \(n=k+1\). Thật vậy, từ \(\left(2\right)\) suy ra:
\(\dfrac{1}{\sqrt{1}}+\dfrac{1}{\sqrt{2}}+...+\dfrac{1}{\sqrt{k}}+\dfrac{1}{\sqrt{k+1}}>\sqrt{k}+\dfrac{1}{\sqrt{k+1}}\)
Vì \(\sqrt{k}+\dfrac{1}{\sqrt{k+1}}=\dfrac{\sqrt{k\left(k+1\right)}+1}{\sqrt{k+1}}>\sqrt{k+1}\)
Nên \(\dfrac{1}{\sqrt{1}}+\dfrac{1}{\sqrt{2}}+...+\dfrac{1}{\sqrt{k}}+\dfrac{1}{\sqrt{k+1}}>\sqrt{k+1}\)
Tức là \(\left(1\right)\) đúng với \(n=k+1\).
Theo nguyên lí quy nạp, (1) đúng với mọi số tự nhiên \(n>1\)
\(A=\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{n\left(n-1\right)}\\ A< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{n-1}-\dfrac{1}{n}=1-\dfrac{1}{n}< 1\left(\dfrac{1}{n}>0\right)\)
Ta có:
\(\dfrac{1}{2^2}< \dfrac{1}{1.2}\)
\(\dfrac{1}{3^2}< \dfrac{1}{2.3}\)
\(\dfrac{1}{4^2}< \dfrac{1}{3.4}\)
...
\(\dfrac{1}{n^2}< \dfrac{1}{n\left(n-1\right)}\)
\(\Rightarrow P< \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{n\left(n-1\right)}\)
\(\Rightarrow P< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{n-1}-\dfrac{1}{n}\)
\(\Rightarrow P< 1-\dfrac{1}{n}< 1\)
\(\Rightarrow P< 1\)
Có: \(Q\left(x\right)=x\left(\dfrac{x^2}{2}-\dfrac{1}{2}x^3+\dfrac{1}{2}x\right)-\left(-\dfrac{1}{2}x^4+x^2\right)\)
\(=\dfrac{x^3}{2}-\dfrac{x^4}{2}+\dfrac{x^2}{2}+\dfrac{x^4}{2}-x^2\)
\(=\dfrac{x^3}{2}-\left(\dfrac{x^4}{2}-\dfrac{x^4}{2}\right)+\left(\dfrac{x^2}{2}-x^2\right)\)
\(=\dfrac{x^3}{2}-\dfrac{x^2}{2}=\dfrac{x^3-x^2}{2}\)
Xét: \(x=2k\left(k\in Z\right)\)
Suy ra: x3 chẵn; x2 chẵn \(\Rightarrow\)x3-x2 chẵn
\(\Rightarrow x^3-x^2⋮2\)
\(\Rightarrow Q\left(x\right)\) nguyên
Xét: \(x=2k+1\left(k\in Z\right)\)
Suy ra: x3 lẻ; x2 lẻ \(\Rightarrow\) x3 - x2 chẵn
\(\Rightarrow x^3-x^2⋮2\)
\(\Rightarrow Q\left(x\right)\) nguyên
Vậy Q(x) luôn nhận giá trị nguyên với mọi số nguyên x
Ta có:\(\dfrac{1}{2^3}< \dfrac{1}{1.2.3};\dfrac{1}{3^3}< \dfrac{1}{2.3.4};\dfrac{1}{4^3}< \dfrac{1}{3.4.5};...;\dfrac{1}{n^3}< \dfrac{1}{\left(n-1\right).n.\left(n+1\right)}\)Vậy:\(\dfrac{1}{2^3}+\dfrac{1}{3^3}+\dfrac{1}{4^3}+...+\dfrac{1}{n^3}< \dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+\dfrac{1}{3.4.5}+...+\dfrac{1}{\left(n-1\right).n.\left(n+1\right)}\)Ta có:\(\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+...+\dfrac{1}{\left(n-1\right).n.\left(n+1\right)}\)
=\(\dfrac{1}{2}.\left(\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+...+\dfrac{1}{\left(n-1\right).n}-\dfrac{1}{n.\left(n+1\right)}\right)\)=\(\dfrac{1}{2}.\left(\dfrac{1}{2}-\dfrac{1}{n.\left(n+1\right)}\right)\)
=\(\dfrac{1}{4}-\dfrac{1}{2n.\left(n+1\right)}< \dfrac{1}{4}\)
Vì:\(\dfrac{1}{2^3}+\dfrac{1}{3^3}+...+\dfrac{1}{n^3}< \dfrac{1}{4}-\dfrac{1}{2n.\left(n+1\right)}< \dfrac{1}{4}\)
\(\Rightarrow\dfrac{1}{2^3}+\dfrac{1}{3^3}+...+\dfrac{1}{n^3}< \dfrac{1}{4}\) hay \(A< \dfrac{1}{4}\)