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

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


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The solution

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  oo         
____         
\   `        
 \    2*n + 1
  \   -------
  /       n  
 /       2   
/___,        
n = 1
$$\sum_{n=1}^{\infty} \frac{2 n + 1}{2^{n}}$$ 
Sum((2*n + 1)/2^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{2 n + 1}{2^{n}}$$
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
$$a_{n} = 2 n + 1$$
and
$$x_{0} = -2$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(\frac{2 n + 1}{2 n + 3}\right)\right)$$
Let's take the limit
we find
False

$$R = 0$$
The rate of convergence of the power series
The answer [src] 
5
$$5$$ 
5
Numerical answer [src] 
5.00000000000000000000000000000
5.00000000000000000000000000000
The graph
Sum of series (2n+1)/2^n

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