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2/(10^(4n+1))
Sum of series/ 2/(10^(4n+1))

Sum of series 2/(10^(4n+1))


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The solution

You have entered [src] 
  oo           
____           
\   `          
 \        2    
  \   ---------
  /     4*n + 1
 /    10       
/___,          
n = 1
$$\sum_{n=1}^{\infty} \frac{2}{10^{4 n + 1}}$$ 
Sum(2/10^(4*n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{2}{10^{4 n + 1}}$$
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
$$a_{n} = 2 \cdot 10^{- 4 n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(10^{- 4 n - 1} \cdot 10^{4 n + 5}\right)$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
The answer [src] 
1/49995
$$\frac{1}{49995}$$ 
1/49995
Numerical answer [src] 
0.0000200020002000200020002000200020
0.0000200020002000200020002000200020
The graph
Sum of series 2/(10^(4n+1))

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