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Sum of series/ (x^n)/n

Sum of series (x^n)/n


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The solution

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

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