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Sum of series/ a^k/k

Sum of series a^k/k


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The solution

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

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