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Question Number 195419 by CrispyXYZ last updated on 02/Aug/23

$$x\ne 0.\underset{i=1}{\sum ^n}{ix}^{i-1}=?$$

Answered by MathedUp last updated on 02/Aug/23

$$x\ne 0.\underset{i=1}{\sum ^n}{ix}^{i-1}=?$$$$\underset{k=1}{\sum ^m}{kx}^{k-1}=\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{\Sigma }{x}^k=\frac{\mathrm{d}}{\mathrm{d}x}\frac{x(x^n-1)}{x-1}$$

Commented by MathedUp last updated on 02/Aug/23

$$\therefore \frac{m\cdot x^{m+1}-(m+1)\cdot x^m+1}{{(x-1)}^2}$$

Commented by MathedUp last updated on 02/Aug/23

$$\mathrm{Bonus}\mathrm{if}x\to 1\underset{k=1}{\sum ^m}k\cdot x^{k-1}=\underset{k=1}{\sum ^m}k$$$$\underset{x\to 1}{\lim }\frac{m\cdot x^{m+1}-(m+1)x^m+1}{{(x-1)}^2}=\frac{m(m+1)}{2}$$$$\mathrm{and}\mathrm{we}\mathrm{can}\mathrm{get}\mathrm{Ramanujan}\mathrm{Sum}\mathrm{by}$$$$\mathrm{Change}\mathrm{this}\mathrm{Equation}$$$$\mathrm{So}\mathrm{Funny}:⟩$$

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