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pi^n+1/2^2n
Sum of series/ pi^n+1/2^2n

Sum of series pi^n+1/2^2n


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The solution

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

False

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+1/2^2n

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