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-1/n^2
Sum of series/ -1/n^2

Sum of series -1/n^2


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
   2 
-pi  
-----
  6
$$- \frac{\pi^{2}}{6}$$ 
-pi^2/6
Numerical answer [src] 
-1.64493406684822643647241516665
-1.64493406684822643647241516665
The graph
Sum of series -1/n^2

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