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((-1)^(n+1))/(n((n+3)^(1/6)))
Sum of series/ ((-1)^(n+1))/(n((n+3)^(1/6)))

Sum of series ((-1)^(n+1))/(n((n+3)^(1/6)))


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The solution

You have entered [src] 
  oo             
____             
\   `            
 \         n + 1 
  \    (-1)      
   )  -----------
  /     6 _______
 /    n*\/ n + 3 
/___,            
n = 1
$$\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n \sqrt[6]{n + 3}}$$ 
Sum((-1)^(n + 1)/((n*(n + 3)^(1/6))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{n + 1}}{n \sqrt[6]{n + 3}}$$
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
$$a_{n} = \frac{\left(-1\right)^{n + 1}}{n \sqrt[6]{n + 3}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \sqrt[6]{n + 4}}{n \sqrt[6]{n + 3}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo             
____             
\   `            
 \         1 + n 
  \    (-1)      
   )  -----------
  /     6 _______
 /    n*\/ 3 + n 
/___,            
n = 1
$$\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n \sqrt[6]{n + 3}}$$ 
Sum((-1)^(1 + n)/(n*(3 + n)^(1/6)), (n, 1, oo))
The graph
Sum of series ((-1)^(n+1))/(n((n+3)^(1/6)))

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