Mister Exam

Other calculators

n(1-cos(pi/n^2))
Sum of series/ n(1-cos(pi/n^2))

Sum of series n(1-cos(pi/n^2))


∑
=

The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
2.96387625930109511160298699125
2.96387625930109511160298699125
The graph
Sum of series n(1-cos(pi/n^2))

    Examples of finding the sum of a series

    © Mister Exam – Calculator