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(n+1)/5^n
Sum of series/ (n+1)/5^n

Sum of series (n+1)/5^n


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The solution

You have entered [src] 
  oo       
____       
\   `      
 \    n + 1
  \   -----
  /      n 
 /      5  
/___,      
n = 1
$$\sum_{n=1}^{\infty} \frac{n + 1}{5^{n}}$$ 
Sum((n + 1)/5^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n + 1}{5^{n}}$$
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
$$a_{n} = n + 1$$
and
$$x_{0} = -5$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-5 + \lim_{n \to \infty}\left(\frac{n + 1}{n + 2}\right)\right)$$
Let's take the limit
we find
False

$$R = 0$$
The rate of convergence of the power series
The answer [src] 
9/16
$$\frac{9}{16}$$ 
9/16
Numerical answer [src] 
0.562500000000000000000000000000
0.562500000000000000000000000000
The graph
Sum of series (n+1)/5^n

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