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Sum of series/ cbrt(x)/(3*x+2)

Sum of series cbrt(x)/(3*x+2)


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The solution

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

False
The answer [src] 
   3 ___
oo*\/ x 
--------
2 + 3*x
$$\frac{\infty \sqrt[3]{x}}{3 x + 2}$$ 
oo*x^(1/3)/(2 + 3*x)

    Examples of finding the sum of a series

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