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# Compute $1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{4} + 3 \cdot \frac {1}{8} + \dots + n \cdot \frac {1}{2^n} + \dotsb.$

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Compute $$1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{4} + 3 \cdot \frac {1}{8} + \dots + n \cdot \frac {1}{2^n} + \dotsb.$$

Apr 12, 2019

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sumfor(n, 1, 1000, n/(2^n) = 2

Apr 12, 2019
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Compute

$$1 \cdot \dfrac {1}{2} + 2 \cdot \dfrac {1}{4} + 3 \cdot \dfrac {1}{8} + \dots + n \cdot \dfrac {1}{2^n} + \dotsb$$

$$\text{Let x = \dfrac{1}{2} \quad|x|<1 }$$

Now we have:  $$\displaystyle s= \sum \limits_{n=1}^{\infty} nx^n$$

Let the progression to be summed be put equal to s:

$$s = x+2x^2+3x^3+4x^4+\cdots + nx^n +\cdots$$

It is divided by $$x$$ and multiplied by $$dx$$ then

$$\dfrac{s\ dx}{x} = dx +2x\ dx+3x^2\ dx+4x^3\ dx+\cdots + nx^{n-1}\ dx +\cdots$$

and with the integrals taken this equation is found
$$\displaystyle \int \dfrac{s\ dx}{x} = x +x^2+x^3+x^4 +\cdots + x^n + \cdots = \dfrac{x}{1-x} \quad \text{ infinite geometric progression }$$

Therefore from the equation:
$$\displaystyle \int \dfrac{s\ dx}{x} = \dfrac{x}{1-x}$$

on differentiation s is found. For the equation becomes:
$$\begin{array}{rcll} \displaystyle \dfrac{s}{x} &=& \dfrac{x}{1-x}\left( \dfrac{1}{x}- \dfrac{(-1)}{1-x} \right) \\\\ &=& \dfrac{1}{1-x} + x(-1)(1-x)^{-2}(-1) \\\\ &=& \dfrac{1}{1-x} + \dfrac{x}{(1-x)^2} \\\\ &=& \dfrac{1-x+x}{(1-x)^2} \\\\ &=& \dfrac{1 }{(1-x)^2} \end{array}$$

thus there is produced:
$$\displaystyle s = \dfrac{x}{(1-x)^2} \\$$

So  $$\text{let x = \dfrac{1}{2} }:$$

$$\begin{array}{|rcll|} \hline s &=& \dfrac{ \dfrac{1}{2} } {\left(1-\dfrac{1}{2} \right)^2} \\\\ &=& \dfrac{ \dfrac{1}{2} } {\left( \dfrac{1}{2} \right)^2} \\\\ &=& \dfrac{ 1 } { \dfrac{1}{2} } \\\\ \mathbf{s} & \mathbf{=} & \mathbf{2} \\ \hline \end{array}$$

finally
$$\displaystyle 1 \cdot \dfrac {1}{2} + 2 \cdot \dfrac {1}{4} + 3 \cdot \dfrac {1}{8} + \dots + n \cdot \dfrac {1}{2^n} + \dotsb = 2$$

Apr 12, 2019