Mister Exam

Other calculators

  • How to use it?

  • Sum of series:
  • 1/((3n-2)(3n+1)) 1/((3n-2)(3n+1))
  • 3/(n(n+2)) 3/(n(n+2))
  • (n+1)/5^n (n+1)/5^n
  • 6/9n^2+12n-5 6/9n^2+12n-5

  • Identical expressions

  • cos(pi*n)/((n^ three + one)^ two)
  • co sinus of e of ( Pi multiply by n) divide by ((n cubed plus 1) squared )
  • co sinus of e of ( Pi multiply by n) divide by ((n to the power of three plus one) to the power of two)
  • cos(pi*n)/((n3+1)2)
  • cospi*n/n3+12
  • cos(pi*n)/((n³+1)²)
  • cos(pi*n)/((n to the power of 3+1) to the power of 2)
  • cos(pin)/((n^3+1)^2)
  • cos(pin)/((n3+1)2)
  • cospin/n3+12
  • cospin/n^3+1^2
  • cos(pi*n) divide by ((n^3+1)^2)

  • Similar expressions

  • cos(pi*n)/((n^3-1)^2)
Sum of series/ cos(pi*n)/((n^3+1)^2)

Sum of series cos(pi*n)/((n^3+1)^2)


∑
=

The solution

You have entered [src] 
  oo           
____           
\   `          
 \    cos(pi*n)
  \   ---------
   )          2
  /   / 3    \ 
 /    \n  + 1/ 
/___,          
i = 1
$$\sum_{i=1}^{\infty} \frac{\cos{\left(\pi n \right)}}{\left(n^{3} + 1\right)^{2}}$$ 
Sum(cos(pi*n)/(n^3 + 1)^2, (i, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\cos{\left(\pi n \right)}}{\left(n^{3} + 1\right)^{2}}$$
It is a series of species
$$a_{i} \left(c x - x_{0}\right)^{d i}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{i \to \infty} \left|{\frac{a_{i}}{a_{i + 1}}}\right|}{c}$$
In this case
$$a_{i} = \frac{\cos{\left(\pi n \right)}}{\left(n^{3} + 1\right)^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{i \to \infty} 1$$
Let's take the limit
we find
True

False
The answer [src] 
oo*cos(pi*n)
------------
         2  
 /     3\   
 \1 + n /
$$\frac{\infty \cos{\left(\pi n \right)}}{\left(n^{3} + 1\right)^{2}}$$ 
oo*cos(pi*n)/(1 + n^3)^2

    Examples of finding the sum of a series

    © Mister Exam – Calculator