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Sum of series/ 1/n^1,5

Sum of series 1/n^1,5

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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)

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