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I want to calculate this sum. How can I do it ? $$ \sum_{i=1}^n i \cdot \left(\frac{9}{10}\right)^i$$ I know how to calc geometric sum, but how to calc this?

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See this. –  Git Gud May 20 at 10:35
    
See a related one: math.stackexchange.com/questions/250746/… –  gar May 20 at 10:35
    
If you take into account the other comments, it is the derivative of a geometric sum. Then ... –  Claude Leibovici May 20 at 10:42

2 Answers 2

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Let x = 0.9

The sum is $\displaystyle x + 2x^2 + 3x^3 + ... + nx^n \\ = x(1 + 2x + 3x^2 + ... nx^{n-1}) \\ = x.\frac{d}{dx}(x + x^2 + x^3 + ... + x^n) \\ = x.\frac{d}{dx}(\frac{x(x^n-1)}{x-1})$

Evaluate the derivative by quotient rule and simplify the expression to give a general sum. Now you can put 0.9 back into it. ;)

Another way: call $\displaystyle x + 2x^2 + 3x^3 + ... + nx^n = S_n(x)\\ xS_n(x) = x^2 + 2x^3 + ... + (n-1)x^n + nx^{n+1}$

Subtracting, $\displaystyle (1-x)S_n(x)= x + x^2 + x^3 + ... + x^n - nx^{n+1} = \frac{x(x^n-1)}{x-1} - nx^{n+1}$

The rest is just algebra. This way you avoid calculus.

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Hint: Differentiate $$\sum_{i=1}^{n}x^i$$ with respect to $x$.

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