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Xét khai triển:
\(\left(1+x\right)^n=C_n^0+C_n^1x+C_n^2x^2+C_n^3x^3+...+C_n^nx^n\)
Đạo hàm 2 vế:
\(n\left(1+x\right)^{n-1}=C_n^1+2C_n^2x+3C_n^3x^2+...+nC_n^nx^{n-1}\)
Thay \(x=1\) và \(n=2017\) vào ta được:
\(2017.2^{2016}=C_{2017^1}+2C_{2017}^2+3C_{2017}^3+...+2017.C_{2017}^{2017}\)
\(S=\dfrac{1}{2018!\left(2019-2018\right)!}+\dfrac{1}{2016!\left(2019-2016\right)!}+...+\dfrac{1}{2!\left(2019-2\right)!}+\dfrac{1}{0!\left(2019-0!\right)}\)
\(\Rightarrow2019!.S=\dfrac{2019!}{2018!\left(2019-2018\right)!}+\dfrac{2019!}{2016!\left(2019-2016\right)!}+...+\dfrac{2019!}{2!\left(2019-2\right)!}+\dfrac{2019!}{0!\left(2019-0\right)!}\)
\(=C_{2019}^{2018}+C_{2019}^{2016}+...+C_{2019}^2+C_{2019}^0\)
\(=\dfrac{1}{2}\left(C_{2019}^0+C_{2019}^1+...+C_{2019}^{2018}+C_{2019}^{2019}\right)\)
\(=\dfrac{1}{2}.2^{2019}=2^{2018}\)
\(\Rightarrow S=\dfrac{2^{2018}}{2019!}\)
Xét khai triển:
\(\left(1+x\right)^{2017}=C_{2017}^0+xC_{2017}^1+x^2C_{2017}^2+...+x^{2017}C_{2017}^{2017}\)
Lấy tích phân 2 vế:
\(\int\limits^1_0\left(1+x\right)^{2017}=\int\limits^1_0\left(C_{2017}^0+xC_{2017}^1+...+x^{2017}C_{2017}^{2017}\right)\)
\(\Leftrightarrow\dfrac{2^{2018}-1}{2018}=C_{2017}^0+\dfrac{1}{2}C_{2017}^1+...+\dfrac{1}{2018}C_{2017}^{2017}\)
Vậy \(S=\dfrac{2^{2018}-1}{2018}\)
\(P=\left[\left(7+4\sqrt{3}\right)\left(4\sqrt{3}-7\right)\right]^{2016}\cdot\left(7+4\sqrt{3}\right)=\left(-1\right)^{2016}\cdot\left(7+4\sqrt{3}\right)=7+4\sqrt{3}\)
\(\lim\limits_{x\rightarrow3}f\left(x\right)=\lim\limits_{x\rightarrow3}\frac{8x^{2016}-24x^{2015}}{x^{2017}+2x^{2016}-15x^{2015}}=\lim\limits_{x\rightarrow3}\frac{8\left(x-3\right)}{x^2+2x-15}=\lim\limits_{x\rightarrow3}\frac{8\left(x-3\right)}{\left(x-3\right)\left(x+5\right)}=\lim\limits_{x\rightarrow3}\frac{8}{x+5}=1\)
\(\lim\limits_{x\rightarrow1}g\left(x\right)=\lim\limits_{x\rightarrow1}\frac{\sqrt{2x+2}-2+2-\sqrt{3x+1}}{m\left(x-1\right)\left(x+1\right)}\)
\(=\lim\limits_{x\rightarrow1}\frac{\frac{2\left(x-1\right)}{\sqrt{2x+2}+2}-\frac{3\left(x-1\right)}{2+\sqrt{3x+1}}}{m\left(x-1\right)\left(x+1\right)}=\lim\limits_{x\rightarrow1}\frac{\frac{2}{\sqrt{2x+2}+2}-\frac{3}{2+\sqrt{3x+1}}}{m\left(x+1\right)}=\frac{\frac{2}{4}-\frac{3}{4}}{2m}=-\frac{1}{8m}\)
\(\Rightarrow-\frac{1}{8m}=1\Rightarrow m=-\frac{1}{8}\)
