<s>[INST] Evaluate $\sum_{n=1}^\infty nx^{n-1}$ How can you evaluate $\sum_{n=1}^\infty nx^{n-1} = \frac{1}{(1-x)^2}$ without relying on the fact that it's the derivative of $\sum_{n=1}^\infty x^n = \frac{1}{1-x} $?
 [/INST] $$a=1+2x+3x^2+4x^3+5x^4+6x^5+...\\|x|<1\\$$multiply a by x $$ xa=x+2x^2+3x^3+4x^4+5x^5+6x^6+...$$now subtract a and ax $$a-xa=1+(2x-x)+(3x^2-2x^2)+(4x^3-3x^3)+...\\a(1-x)=1+x+x^2+x^3+x^4+x^5+...\\ $$$$ a(1-x)=\frac{1}{1-x}\\a=\frac{1}{(1-x)^2}$$
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