Sum of series x*a^x

The solution

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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

The Sum of the Power Series
```
x^n/n
```
```
(x-1)^n
```
Factorial
```
1/2^(n!)
```
```
n^2/n!
```
```
x^n/n!
```
```
k!/(n!*(n+k)!)
```
Flint Hills Series
```
csc(n)^2/n^3
```
Basel problem
```
1/n^2
```
```
1/n^4
```
```
1/n^6
```
Harmonic series
```
1/n
```
Grandi's series
```
(-1)^n
```
Alternating series
```
(-1)^(n + 1)/n
```
```
(n + 2)*(-1)^(n - 1)
```
```
(3*n - 1)/(-5)^n
```
```
(-1)^(n - 1)*n/(6*n - 5)
```
Newton–Mercator series
```
(-1)^(n + 1)/n*x^n
```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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Identical expressions

Sum of series x*a^x

The solution

Examples of finding the sum of a series