Mister Exam

Other calculators

Sum of series/ 1/x^n

Sum of series 1/x^n


∑
=

The solution

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

    Examples of finding the sum of a series

    © Mister Exam – Calculator