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pi^(-n)/pi^n+3
Sum of series/ pi^(-n)/pi^n+3

Sum of series pi^(-n)/pi^n+3


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The solution

You have entered [src] 
  oo            
____            
\   `           
 \    /  -n    \
  \   |pi      |
   )  |---- + 3|
  /   |  n     |
 /    \pi      /
/___,           
n = 1
$$\sum_{n=1}^{\infty} \left(3 + \frac{\pi^{- n}}{\pi^{n}}\right)$$ 
Sum(pi^(-n)/pi^n + 3, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$3 + \frac{\pi^{- n}}{\pi^{n}}$$
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
$$a_{n} = 3 + \pi^{- 2 n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{3 + \pi^{- 2 n}}{3 + \pi^{- (2 n + 2)}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
oo
$$\infty$$ 
oo
Numerical answer
The series diverges
The graph
Sum of series pi^(-n)/pi^n+3

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