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sqrt(n-1)/(n^2+1)*arctg(n+1)/sqrt^3(n^4+5)

  • How to use it?

  • Sum of series:
  • 2i 2i
  • 1/(n+1)! 1/(n+1)!
  • lnn/n lnn/n
  • factorial(n+2)/n^n factorial(n+2)/n^n

  • Identical expressions

  • sqrt(n- one)/(n^ two + one)*arctg(n+ one)/sqrt^ three (n^ four + five)
  • square root of (n minus 1) divide by (n squared plus 1) multiply by arctg(n plus 1) divide by square root of cubed (n to the power of 4 plus 5)
  • square root of (n minus one) divide by (n to the power of two plus one) multiply by arctg(n plus one) divide by square root of to the power of three (n to the power of four plus five)
  • √(n-1)/(n^2+1)*arctg(n+1)/√^3(n^4+5)
  • sqrt(n-1)/(n2+1)*arctg(n+1)/sqrt3(n4+5)
  • sqrtn-1/n2+1*arctgn+1/sqrt3n4+5
  • sqrt(n-1)/(n²+1)*arctg(n+1)/sqrt³(n⁴+5)
  • sqrt(n-1)/(n to the power of 2+1)*arctg(n+1)/sqrt to the power of 3(n to the power of 4+5)
  • sqrt(n-1)/(n^2+1)arctg(n+1)/sqrt^3(n^4+5)
  • sqrt(n-1)/(n2+1)arctg(n+1)/sqrt3(n4+5)
  • sqrtn-1/n2+1arctgn+1/sqrt3n4+5
  • sqrtn-1/n^2+1arctgn+1/sqrt^3n^4+5
  • sqrt(n-1) divide by (n^2+1)*arctg(n+1) divide by sqrt^3(n^4+5)

  • Similar expressions

  • sqrt(n+1)/(n^2+1)*arctg(n+1)/sqrt^3(n^4+5)
  • sqrt(n-1)/(n^2-1)*arctg(n+1)/sqrt^3(n^4+5)
  • sqrt(n-1)/(n^2+1)*arctg(n+1)/sqrt^3(n^4-5)
  • sqrt(n-1)/(n^2+1)*arctg(n-1)/sqrt^3(n^4+5)
Sum of series/ sqrt(n-1)/(n^2+1)*arctg(n+1)/sqrt^3(n^4+5)

Sum of series sqrt(n-1)/(n^2+1)*arctg(n+1)/sqrt^3(n^4+5)


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The solution

You have entered [src] 
   oo                         
_______                       
\      `                      
 \         _______            
  \      \/ n - 1             
   \     ---------*atan(n + 1)
    \       2                 
     \     n  + 1             
     /   ---------------------
    /                    3    
   /             ________     
  /             /  4          
 /            \/  n  + 5      
/______,                      
 n = 3
$$\sum_{n=3}^{\infty} \frac{\frac{\sqrt{n - 1}}{n^{2} + 1} \operatorname{atan}{\left(n + 1 \right)}}{\left(\sqrt{n^{4} + 5}\right)^{3}}$$ 
Sum(((sqrt(n - 1)/(n^2 + 1))*atan(n + 1))/(sqrt(n^4 + 5))^3, (n, 3, oo))
The radius of convergence of the power series
Given number:
$$\frac{\frac{\sqrt{n - 1}}{n^{2} + 1} \operatorname{atan}{\left(n + 1 \right)}}{\left(\sqrt{n^{4} + 5}\right)^{3}}$$
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
$$a_{n} = \frac{\sqrt{n - 1} \operatorname{atan}{\left(n + 1 \right)}}{\left(n^{2} + 1\right) \left(n^{4} + 5\right)^{\frac{3}{2}}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(\left(n + 1\right)^{2} + 1\right) \left(\left(n + 1\right)^{4} + 5\right)^{\frac{3}{2}} \left|{\sqrt{n - 1}}\right| \operatorname{atan}{\left(n + 1 \right)}}{\sqrt{n} \left(n^{2} + 1\right) \left(n^{4} + 5\right)^{\frac{3}{2}} \operatorname{atan}{\left(n + 2 \right)}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo                         
_____                        
\    `                       
 \       ________            
  \    \/ -1 + n *atan(1 + n)
   \   ----------------------
   /                     3/2 
  /     /     2\ /     4\    
 /      \1 + n /*\5 + n /    
/____,                       
n = 3
$$\sum_{n=3}^{\infty} \frac{\sqrt{n - 1} \operatorname{atan}{\left(n + 1 \right)}}{\left(n^{2} + 1\right) \left(n^{4} + 5\right)^{\frac{3}{2}}}$$ 
Sum(sqrt(-1 + n)*atan(1 + n)/((1 + n^2)*(5 + n^4)^(3/2)), (n, 3, oo))
Numerical answer [src] 
0.000277978305947414188349605122489
0.000277978305947414188349605122489
The graph
Sum of series sqrt(n-1)/(n^2+1)*arctg(n+1)/sqrt^3(n^4+5)

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