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sqrt((n+4)/((n^4)+4))
Sum of series/ sqrt((n+4)/((n^4)+4))

Sum of series sqrt((n+4)/((n^4)+4))


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
  oo              
_____             
\    `            
 \        _______ 
  \     \/ 4 + n  
   \   -----------
   /      ________
  /      /      4 
 /     \/  4 + n  
/____,            
n = 1
$$\sum_{n=1}^{\infty} \frac{\sqrt{n + 4}}{\sqrt{n^{4} + 4}}$$ 
Sum(sqrt(4 + n)/sqrt(4 + n^4), (n, 1, oo))
The graph
Sum of series sqrt((n+4)/((n^4)+4))

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