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cos(npi*2)/n^4
  • How to use it?

  • Sum of series:
  • cos(npi*2)/n^4 cos(npi*2)/n^4
  • sin2x
  • (2*n-1)/2^n (2*n-1)/2^n
  • (-1)^n (-1)^n
  • Identical expressions

  • cos(npi* two)/n^ four
  • co sinus of e of (n Pi multiply by 2) divide by n to the power of 4
  • co sinus of e of (n Pi multiply by two) divide by n to the power of four
  • cos(npi*2)/n4
  • cosnpi*2/n4
  • cos(npi*2)/n⁴
  • cos(npi2)/n^4
  • cos(npi2)/n4
  • cosnpi2/n4
  • cosnpi2/n^4
  • cos(npi*2) divide by n^4

Sum of series cos(npi*2)/n^4



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The solution

You have entered [src]
  oo             
____             
\   `            
 \    cos(n*pi*2)
  \   -----------
  /         4    
 /         n     
/___,            
n = 1            
n=1cos(2πn)n4\sum_{n=1}^{\infty} \frac{\cos{\left(2 \pi n \right)}}{n^{4}}
Sum(cos((n*pi)*2)/n^4, (n, 1, oo))
The radius of convergence of the power series
Given number:
cos(2πn)n4\frac{\cos{\left(2 \pi n \right)}}{n^{4}}
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=cos(2πn)n4a_{n} = \frac{\cos{\left(2 \pi n \right)}}{n^{4}}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limn((n+1)4cos(2πn)cos(π(2n+2))n4)1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{4} \left|{\frac{\cos{\left(2 \pi n \right)}}{\cos{\left(\pi \left(2 n + 2\right) \right)}}}\right|}{n^{4}}\right)
Let's take the limit
we find
True

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50.901.10
The answer [src]
  oo             
____             
\   `            
 \    cos(2*pi*n)
  \   -----------
  /         4    
 /         n     
/___,            
n = 1            
n=1cos(2πn)n4\sum_{n=1}^{\infty} \frac{\cos{\left(2 \pi n \right)}}{n^{4}}
Sum(cos(2*pi*n)/n^4, (n, 1, oo))
The graph
Sum of series cos(npi*2)/n^4

    Examples of finding the sum of a series