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Sum of series/ (6x-1)^n/((n+1)*6^n)

Sum of series (6x-1)^n/((n+1)*6^n)


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The solution

You have entered [src] 
  oo            
____            
\   `           
 \             n
  \   (6*x - 1) 
   )  ----------
  /            n
 /    (n + 1)*6 
/___,           
n = 1
$$\sum_{n=1}^{\infty} \frac{\left(6 x - 1\right)^{n}}{6^{n} \left(n + 1\right)}$$ 
Sum((6*x - 1)^n/(((n + 1)*6^n)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(6 x - 1\right)^{n}}{6^{n} \left(n + 1\right)}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{6^{- n}}{n + 1}$$
and
$$x_{0} = 1$$
,
$$d = 1$$
,
$$c = 6$$
then
$$R = \frac{1 + \lim_{n \to \infty}\left(\frac{6^{- n} 6^{n + 1} \left(n + 2\right)}{n + 1}\right)}{6}$$
Let's take the limit
we find
$$R = \frac{7}{6}$$
The answer [src] 
//  1    x\ /     2       2*log(7/6 - x)\                             
||- -- + -|*|- -------- - --------------|  for And(x >= -5/6, x < 7/6)
|\  12   2/ |  -1/6 + x              2  |                             
|           \              (-1/6 + x)   /                             
|                                                                     
|           oo                                                        
|         ____                                                        
<         \   `                                                       
|          \     -n           n                                       
|           \   6  *(-1 + 6*x)                                        
|           /   ---------------                     otherwise         
|          /         1 + n                                            
|         /___,                                                       
|         n = 1                                                       
\
$$\begin{cases} \left(\frac{x}{2} - \frac{1}{12}\right) \left(- \frac{2}{x - \frac{1}{6}} - \frac{2 \log{\left(\frac{7}{6} - x \right)}}{\left(x - \frac{1}{6}\right)^{2}}\right) & \text{for}\: x \geq - \frac{5}{6} \wedge x < \frac{7}{6} \\\sum_{n=1}^{\infty} \frac{6^{- n} \left(6 x - 1\right)^{n}}{n + 1} & \text{otherwise} \end{cases}$$ 
Piecewise(((-1/12 + x/2)*(-2/(-1/6 + x) - 2*log(7/6 - x)/(-1/6 + x)^2), (x >= -5/6)∧(x < 7/6)), (Sum(6^(-n)*(-1 + 6*x)^n/(1 + n), (n, 1, oo)), True))

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