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Xét 2 khai triển:
\(\left(x+1\right)^{2018}=C_{2018}^0+C_{2018}^1x+C_{2018}^2x^2+...+C_{2018}^{2018}x^{2018}\)
\(\left(x-1\right)^{2018}=C_{2018}^0-C_{2018}^1x+C_{2018}^2x^2-...+C_{2018}^{2018}x^{2018}\)
Cộng vế với vế:
\(\left(x+1\right)^{2018}+\left(x-1\right)^{2018}=2\left(C_{2018}^0+C_{2018}^2x^2+...+C_{2018}^{2018}x^{2018}\right)\)
\(\Leftrightarrow C_{2018}^0+C_{2018}^2x^2+...+C_{2018}^{2018}x^{2018}=\frac{1}{2}\left(x+1\right)^{2018}+\frac{1}{2}\left(x-1\right)^{2018}\)
\(\Rightarrow\lim\limits_{x\rightarrow1}=\frac{\frac{1}{2}\left(x+1\right)^{2018}+\frac{1}{2}\left(x-1\right)^{2018}-2^{2017}}{x-1}=\lim\limits_{x\rightarrow1}\frac{1009\left(x+1\right)^{2017}+1009\left(x-1\right)^{2017}}{1}=1009.2^{2017}\)
Lời giải:
\(\frac{(x^2+x+1)^{2018}+(x+2)^{2018}-2.3^{2018}}{(x-1)(x+2017)}=\frac{(x^2+x+1)^{2018}-3^{2018}+(x+2)^{2018}-3^{2018}}{(x-1)(x+2017)}\)
\(=\frac{(x^2+x-2)[(x^2+x+1)^{2017}+...+3^{2017}]+(x-1)[(x+2)^{2017}+...+3^{2017}]}{(x-1)(x+2017)}\)
\(=\frac{(x+2)[(x^2+x+1)^{2017}+...+3^{2017}]+(x+2)^{2017}+...+3^{2017}}{x+2017}\)
Do đó:
\(\lim_{x\to 1}\frac{(x^2+x+1)^{2018}+(x+2)^{2018}-2.3^{2018}}{(x-1)(x+2017)}=\lim_{x\to 1}\frac{(x+2)[(x^2+x+1)^{2017}+...+3^{2017}]+(x+2)^{2017}+...+3^{2017}}{x+2017}\)
\(=\frac{3\underbrace{(3^{2017}+3^{2017}+...+3^{2017})}_{2018}+\underbrace{3^{2017}+...+3^{2017}}_{2018}}{1+2017}\)
\(=\frac{3.2018.3^{2017}+2018.3^{2017}}{2018}=3^{2018}+3^{2017}=3^{2017}.4\)
Lim(-2n^2019+3n^2018+4)
=Lim n^2019(-2+3/n+4/n^2019)
=Âm vô cực
Lời giải:
Đặt \(\underbrace{11...1}_{n}=a\Rightarrow 9a+1=10^n\Rightarrow a=\frac{10^n-1}{9}\Rightarrow \underbrace{44...4}_{n}=4a=\frac{4}{9}(10^n-1)\)
Thay $n=1,2,...,2018$ và đặt tổng cần tính là $T$
Khi đó:
\(T=\frac{4}{9}(10^1-1)+\frac{4}{9}(10^2-1)+\frac{4}{9}(10^3-1)+...+\frac{4}{9}(10^{2018}-1)\)
\(=\frac{4}{9}(10+10^2+10^3+...+10^{2018}-2018)\)
\(10T=\frac{4}{9}(10^2+10^3+...+10^{2019}-20180)\)
Trừ theo vế:
\(9T=10T-T=\frac{4}{9}(10^{2019}-20180-10+2018)=\frac{4}{9}(10^{2019}-18172)\)
\(\Rightarrow T=\frac{4(10^{2019}-18172)}{81}\)