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

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


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
  oo              
____              
\   `             
 \         3/2    
  \       n       
   )  ------------
  /              2
 /    1 + n + 2*n 
/___,             
n = 1
$$\sum_{n=1}^{\infty} \frac{n^{\frac{3}{2}}}{2 n^{2} + n + 1}$$ 
Sum(n^(3/2)/(1 + n + 2*n^2), (n, 1, oo))
The graph
Sum of series (nsqrt(n))/(2n^2+n+1)

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