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(n-1)/2^(n+1)
Sum of series/ (n-1)/2^(n+1)

Sum of series (n-1)/2^(n+1)


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The solution

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  oo        
____        
\   `       
 \    n - 1 
  \   ------
  /    n + 1
 /    2     
/___,       
n = 1
$$\sum_{n=1}^{\infty} \frac{n - 1}{2^{n + 1}}$$ 
Sum((n - 1)/2^(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n - 1}{2^{n + 1}}$$
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
$$a_{n} = 2^{- n - 1} \left(n - 1\right)$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{2^{- n - 1} \cdot 2^{n + 2} \left|{n - 1}\right|}{n}\right)$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
The answer [src] 
1/2
$$\frac{1}{2}$$ 
1/2
Numerical answer [src] 
0.500000000000000000000000000000
0.500000000000000000000000000000
The graph
Sum of series (n-1)/2^(n+1)

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