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  • Sum of series:
  • 2 2
  • n^n/factorial(n+3) n^n/factorial(n+3)
  • cos(nx)/n^2
  • 11 11

  • Identical expressions

  • cos(nx)/n^ two
  • co sinus of e of (nx) divide by n squared
  • co sinus of e of (nx) divide by n to the power of two
  • cos(nx)/n2
  • cosnx/n2
  • cos(nx)/n²
  • cos(nx)/n to the power of 2
  • cosnx/n^2
  • cos(nx) divide by n^2
Sum of series/ cos(nx)/n^2

Sum of series cos(nx)/n^2


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The solution

You have entered [src] 
  oo          
____          
\   `         
 \    cos(n*x)
  \   --------
  /       2   
 /       n    
/___,         
n = 1
∑n=1∞cos⁡(nx)n2\sum_{n=1}^{\infty} \frac{\cos{\left(n x \right)}}{n^{2}} 
Sum(cos(n*x)/n^2, (n, 1, oo))
The radius of convergence of the power series
Given number:
cos⁡(nx)n2\frac{\cos{\left(n x \right)}}{n^{2}}
It is a series of species
an(cx−x0)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+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=cos⁡(nx)n2a_{n} = \frac{\cos{\left(n x \right)}}{n^{2}}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞((n+1)2∣cos⁡(nx)cos⁡(x(n+1))∣n2)1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2} \left|{\frac{\cos{\left(n x \right)}}{\cos{\left(x \left(n + 1\right) \right)}}}\right|}{n^{2}}\right)
Let's take the limit
we find
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