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5/n^2

Sum of series 5/n^2



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The solution

You have entered [src]
  oo    
____    
\   `   
 \    5 
  \   --
  /    2
 /    n 
/___,   
n = 1   
n=15n2\sum_{n=1}^{\infty} \frac{5}{n^{2}}
Sum(5/n^2, (n, 1, oo))
The radius of convergence of the power series
Given number:
5n2\frac{5}{n^{2}}
It is a series of species
an(cxx0)dna_{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:
Rd=x0+limnanan+1cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=5n2a_{n} = \frac{5}{n^{2}}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limn((n+1)2n2)1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2}}{n^{2}}\right)
Let's take the limit
we find
True

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.52.510.0
The answer [src]
    2
5*pi 
-----
  6  
5π26\frac{5 \pi^{2}}{6}
5*pi^2/6
Numerical answer [src]
8.22467033424113218236207583323
8.22467033424113218236207583323
The graph
Sum of series 5/n^2

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