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pi^(-n)/pi^n
Sum of series/ pi^(-n)/pi^n

Sum of series pi^(-n)/pi^n


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The solution

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

$$R = 0$$
The rate of convergence of the power series
The answer [src] 
      1      
-------------
  2 /     1 \
pi *|1 - ---|
    |      2|
    \    pi /
$$\frac{1}{\pi^{2} \left(1 - \frac{1}{\pi^{2}}\right)}$$ 
1/(pi^2*(1 - 1/pi^2))
Numerical answer [src] 
0.112744599959518002663912751139
0.112744599959518002663912751139
The graph
Sum of series pi^(-n)/pi^n

    Examples of finding the sum of a series

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