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C=\(\dfrac{x-x^3}{x^2+1}\left(\dfrac{1}{1+2x+x^2}+\dfrac{1}{1-x^2}\right)+\dfrac{1}{1+x}\)
\(=\dfrac{x\left(1-x^2\right)}{x^2+1}\left(\dfrac{1}{\left(1+x\right)^2}+\dfrac{1}{\left(1-x\right)\left(1+x\right)}\right)+\dfrac{1}{1+x}\)
\(=\dfrac{x\left(1-x\right)\left(1+x\right)}{x^2+1}\left(\dfrac{1-x+1+x}{\left(1-x\right)\left(1+x\right)^2}\right)+\dfrac{1}{1+x}\)
\(=\dfrac{x\left(1-x\right)\left(1+x\right).2}{\left(x^2+1\right)\left(1-x\right)\left(1+x^2\right)}+\dfrac{1}{1+x}\)
\(=\dfrac{2x}{\left(x^2+1\right)\left(1+x\right)}+\dfrac{1}{1+x}\)
\(=\dfrac{2x+\left(x^2+1\right)}{\left(x^2+1\right)\left(1+x\right)}\)
\(=\dfrac{2x+x^2+1}{\left(x^2+1\right)\left(x+1\right)}\)
\(=\dfrac{x^2+2x+1}{\left(x^2+1\right)\left(x+1\right)}\)
\(=\dfrac{\left(x+1\right)^2}{\left(x^2+1\right)\left(x +1\right)}\)
\(=\dfrac{x+1}{x^2+1}\)
Lời giải:
Xét một thừa số tổng quát:
\(1-\frac{1}{1+2+...+n}=1-\frac{1}{\frac{n(n+1)}{2}}=1-\frac{2}{n(n+1)}\)
\(1-\frac{1}{1+2+...+n}=\frac{n^2+n-2}{n(n+1)}=\frac{(n-1)(n+2)}{n(n+1)}\)
Do đó:
\(P_n=\left(1-\frac{1}{1+2}\right)\left(1-\frac{1}{1+2+3}\right)....\left(1-\frac{1}{1+2+...+n}\right)\)
\(P_n=\frac{1.4}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}....\frac{(n-1)(n+2)}{n(n+1)}\)
\(P_n=\frac{(1.2.3...(n-1))(4.5.6...(n+2))}{(2.3.4...n)(3.4.5..(n+1))}\)
\(P_n=\frac{1}{n}.\frac{n+2}{3}=\frac{n+2}{3n}\Rightarrow \frac{1}{P_n}=\frac{3n}{n+2}\)
Để \(\frac{1}{P_{n}}\in\mathbb{N}\Rightarrow \frac{3n}{n+2}\in\mathbb{N}\)
\(\Leftrightarrow 3n\vdots n+2\)
\(\Leftrightarrow 3(n+2)-6\vdots n+2\)
\(\Leftrightarrow 6\vdots n+2\)
\(\Rightarrow n+2=6\) do \(n+2>3\forall n>1\)
\(\Leftrightarrow n=4\)
Vậy \(n=4\)