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Sum of series/ x^2/k^3/2

Sum of series x^2/k^3/2


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The solution

You have entered [src] 
  oo       
_____      
\    `     
 \     / 2\
  \    |x |
   \   |--|
    )  | 3|
   /   \k /
  /    ----
 /      2  
/____,     
k = 1
$$\sum_{k=1}^{\infty} \frac{x^{2} \frac{1}{k^{3}}}{2}$$ 
Sum((x^2/k^3)/2, (k, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{x^{2} \frac{1}{k^{3}}}{2}$$
It is a series of species
$$a_{k} \left(c x - x_{0}\right)^{d k}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{k \to \infty} \left|{\frac{a_{k}}{a_{k + 1}}}\right|}{c}$$
In this case
$$a_{k} = \frac{x^{2}}{2 k^{3}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{k \to \infty}\left(\frac{\left(k + 1\right)^{3}}{k^{3}}\right)$$
Let's take the limit
we find
True

False
The answer [src] 
 2        
x *zeta(3)
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    2
$$\frac{x^{2} \zeta\left(3\right)}{2}$$ 
x^2*zeta(3)/2

    Examples of finding the sum of a series

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