Chứng minh $x+2{{x}^{2}}+3{{x}^{3}}+...+n{{x}^{n}}+...=\frac{x}{{{\left( x-1 \right)}^{2}}}$
Edited by phathuy, 23-08-2014 - 17:08.
Chứng minh $x+2{{x}^{2}}+3{{x}^{3}}+...+n{{x}^{n}}+...=\frac{x}{{{\left( x-1 \right)}^{2}}}$
Edited by phathuy, 23-08-2014 - 17:08.
Mục đích của cuộc sống là sống có mục đích
Chứng minh $x+2{{x}^{2}}+3{{x}^{3}}+...+n{{x}^{n}}+...=\frac{x}{{{\left( x-1 \right)}^{2}}}$
Theo công thức khai triển Maclaurin ta có
$$\frac{1}{1-x}=\sum\limits_{n=0}^{\infty}x^n=1+x+x^2+x^3+...$$
$$\frac{1}{(1-x)^2}=1+2x+3x^2+...$$
$$\Rightarrow \frac{x}{(x-1)^2}=x+2x^2+3x^3+... (\text{Đpcm})$$
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