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(n+cosn)/((n^2)(sqrt(n)))
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  • Similar expressions

  • (n-cosn)/((n^2)(sqrt(n)))

Sum of series (n+cosn)/((n^2)(sqrt(n)))



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The solution

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

False
The rate of convergence of the power series
The answer [src]
  oo            
____            
\   `           
 \    n + cos(n)
  \   ----------
  /       5/2   
 /       n      
/___,           
n = 1           
$$\sum_{n=1}^{\infty} \frac{n + \cos{\left(n \right)}}{n^{\frac{5}{2}}}$$
Sum((n + cos(n))/n^(5/2), (n, 1, oo))
The graph
Sum of series (n+cosn)/((n^2)(sqrt(n)))

    Examples of finding the sum of a series