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

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


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The solution

You have entered [src] 
  oo          
____          
\   `         
 \       1    
  \   --------
  /          4
 /    (n + 2) 
/___,         
n = 1
$$\sum_{n=1}^{\infty} \frac{1}{\left(n + 2\right)^{4}}$$ 
Sum(1/((n + 2)^4), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(n + 2\right)^{4}}$$
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
$$a_{n} = \frac{1}{\left(n + 2\right)^{4}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 3\right)^{4}}{\left(n + 2\right)^{4}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
         4
  17   pi 
- -- + ---
  16    90
$$- \frac{17}{16} + \frac{\pi^{4}}{90}$$ 
-17/16 + pi^4/90
Numerical answer [src] 
0.0198232337111381915160036965412
0.0198232337111381915160036965412
The graph
Sum of series 1/(n+2)^4

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