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Sum of series/ x*a^x

Sum of series x*a^x


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The solution

You have entered [src] 
  oo      
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 \  `     
  \      x
  /   x*a 
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x = 1
$$\sum_{x=1}^{\infty} a^{x} x$$ 
Sum(x*a^x, (x, 1, oo))
The radius of convergence of the power series
Given number:
$$a^{x} x$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = x$$
and
$$x_{0} = - a$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(- a + \lim_{x \to \infty}\left(\frac{x}{x + 1}\right)\right)$$
Let's take the limit
we find
$$R = \tilde{\infty} \left(1 - a\right)$$
The answer [src] 
/    a                  
| --------   for |a| < 1
|        2              
| (1 - a)               
|                       
|  oo                   
< ___                   
| \  `                  
|  \      x             
|  /   x*a    otherwise 
| /__,                  
|x = 1                  
\
$$\begin{cases} \frac{a}{\left(1 - a\right)^{2}} & \text{for}\: \left|{a}\right| < 1 \\\sum_{x=1}^{\infty} a^{x} x & \text{otherwise} \end{cases}$$ 
Piecewise((a/(1 - a)^2, |a| < 1), (Sum(x*a^x, (x, 1, oo)), True))

    Examples of finding the sum of a series

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