Lời giải:
\(\frac{n-1}{n!}=\frac{n}{n!}-\frac{1}{n!}=\frac{1}{(n-1)!}-\frac{1}{n!}\). Do đó:
\(\text{VT}=\frac{1}{1!}-\frac{1}{2!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-....+\frac{1}{(n-1)!}-\frac{1}{n!}=1-\frac{1}{n!}< 1\)
Ta có đpcm.
CMR, ∀n ≥ 1, n ∈ N : \(\dfrac{1}{2}\)+\(\dfrac{1}{3\sqrt{2}}\)+\(\dfrac{1}{4\sqrt{3}}\)+....+ \(\dfrac{1}{\left(n+1\right)\sqrt{n}}\)<2
Cho n \(\in\) N; n \(\ge\) 2. CMR:
\(\sqrt{n}< \dfrac{1}{\sqrt{1}}+\dfrac{1}{\sqrt{2}}+.....+\dfrac{1}{\sqrt{n}}< 2\sqrt{n}\)
1) Chứng minh rằng: \(1+\dfrac{1}{2\sqrt{2}}+\dfrac{1}{3\sqrt{3}}+...+\dfrac{1}{n\sqrt{n}}< 2\sqrt{2}\left(n\in N\right)\)
2) Chứng minh rằng: \(\dfrac{2}{3}+\sqrt{n+1}< 1+\sqrt{2}+\sqrt{3}+...+\sqrt{n}< \dfrac{2}{3}\left(n+1\right)\sqrt{n}\)
3) \(2\sqrt{n}-3< \dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{n}}< 2\sqrt{n}-2\)
4) \(\dfrac{\sqrt{2}-\sqrt{1}}{2+1}+\dfrac{\sqrt{3}-\sqrt{2}}{3+2}+...+\dfrac{\sqrt{n+1}-\sqrt{n}}{n+1+n}< \dfrac{1}{2}\left(1-\dfrac{1}{\sqrt{n+1}}\right)\)
CMR:
\(\dfrac{1}{2\sqrt{1}+1\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+\dfrac{1}{4\sqrt{3}+3\sqrt{3}}+....+\dfrac{1}{\left(n+1\right)\left(\sqrt{n}+n\sqrt{n+1}\right)}< 1\)
với n số nguyên dương lớn hơn 1
a) cmr \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< 2-\dfrac{1}{n}\)
b)cmr \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< \dfrac{5}{3}\)
với số nguyên dương lớn hơn 1
a)cmr \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< 2-\dfrac{1}{n}\)
b)cmr \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< \dfrac{5}{3}\)
Cho a, b, c là độ dài 3 cạnh tam giác. CMR:
1, \(\dfrac{1}{\left(a+b-c\right)^n}+\dfrac{1}{\left(a-b+c\right)^n}+\dfrac{1}{\left(b+c-a\right)^n}\ge\dfrac{1}{a^n}+\dfrac{1}{b^n}+\dfrac{1}{c^n}\)
2, \(\dfrac{1}{a^n}+\dfrac{1}{b^n}+\dfrac{1}{c^n}\ge4^n\left[\dfrac{1}{\left(2a+b+c\right)^n}+\dfrac{1}{\left(a+2b+c\right)^n}+\dfrac{1}{\left(a+b+2c\right)^n}\right]\)
Cho các số dương a,b,c thỏa mãn ab + bc + ca = 3. CMR:
\(\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\ge\dfrac{3}{2}\)
cho \(m,n\in Z\) sao cho
\(\dfrac{m}{n}=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...-\dfrac{1}{426}+\dfrac{1}{427}\)
cmr:\(m⋮641\)
