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Question Number 181207 by lapache last updated on 23/Nov/22

$$cacul$$$$\mathrm{\forall }x\in ]0,1[$$$$\underset{n=0}{\sum ^{+\mathrm{\infty }}}{nx}^n$$

Answered by ARUNG_Brandon_MBU last updated on 23/Nov/22

$$S=x+2x^2+3x^3+4x^4+\cdot \cdot \cdot$$$$xS=x^2+2x^3+3x^4+\cdot \cdot \cdot$$$$S-xS=x+x^2+x^3+x^4+\cdot \cdot \cdot =\frac{x}{1-x}$$$$(1-x)S=\frac{x}{1-x}\Rightarrow S=\frac{x}{{(1-x)}^2}$$

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