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

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

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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)

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