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ln(n)/sqrt(n^5+n)

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  • Similar expressions

  • ln(n)/sqrt(n^5-n)
Sum of series/ ln(n)/sqrt(n^5+n)

Sum of series ln(n)/sqrt(n^5+n)


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The solution

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

False
The rate of convergence of the power series
The graph
Sum of series ln(n)/sqrt(n^5+n)

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