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1/(10^(4n+1))
Sum of series/ 1/(10^(4n+1))

Sum of series 1/(10^(4n+1))


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The solution

You have entered [src] 
  oo           
____           
\   `          
 \        1    
  \   ---------
  /     4*n + 1
 /    10       
/___,          
n = 0
∑n=0∞1104n+1\sum_{n=0}^{\infty} \frac{1}{10^{4 n + 1}} 
Sum(1/(10^(4*n + 1)), (n, 0, oo))
The radius of convergence of the power series
Given number:
1104n+1\frac{1}{10^{4 n + 1}}
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=10−4n−1a_{n} = 10^{- 4 n - 1}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞(10−4n−1⋅104n+5)1 = \lim_{n \to \infty}\left(10^{- 4 n - 1} \cdot 10^{4 n + 5}\right)
Let's take the limit
we find
False

False
The rate of convergence of the power series
0.06.00.51.01.52.02.53.03.54.04.55.05.50.099990.10002
The answer [src] 
1000
----
9999
10009999\frac{1000}{9999} 
1000/9999
Numerical answer [src] 
0.100010001000100010001000100010
0.100010001000100010001000100010
The graph
Sum of series 1/(10^(4n+1))

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