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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

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  oo       
____       
\   `      
 \    n + 1
  \   -----
  /      n 
 /      5  
/___,      
n = 1
∑n=1∞n+15n\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:
n+15n\frac{n + 1}{5^{n}}
It is a series of species
an(cx−x0)dna_{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:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=n+1a_{n} = n + 1
and
x0=−5x_{0} = -5
,
d=−1d = -1
,
c=0c = 0
then
1R=∞~(−5+lim⁡n→∞(n+1n+2))\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=0R = 0
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50.20.6
The answer [src] 
9/16
916\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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