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1/n^1,5

Sum of series 1/n^1,5



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The solution

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

False
The rate of convergence of the power series
The answer [src]
zeta(3/2)
$$\zeta\left(\frac{3}{2}\right)$$
zeta(3/2)
Numerical answer [src]
2.61237534868548834334789184423
2.61237534868548834334789184423
The graph
Sum of series 1/n^1,5

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