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cos(2n)/(2^n)

  • How to use it?

  • Sum of series:
  • x*a^x
  • exp(-n) exp(-n)
  • 21 21
  • pi^(2*n)/factorial(2*n) pi^(2*n)/factorial(2*n)

  • Identical expressions

  • cos(two n)/(2^n)
  • co sinus of e of (2n) divide by (2 to the power of n)
  • co sinus of e of (two n) divide by (2 to the power of n)
  • cos(2n)/(2n)
  • cos2n/2n
  • cos2n/2^n
  • cos(2n) divide by (2^n)
Sum of series/ cos(2n)/(2^n)

Sum of series cos(2n)/(2^n)


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The solution

You have entered [src] 
  oo          
____          
\   `         
 \    cos(2*n)
  \   --------
  /       n   
 /       2    
/___,         
n = 1
∑n=1∞cos⁡(2n)2n\sum_{n=1}^{\infty} \frac{\cos{\left(2 n \right)}}{2^{n}} 
Sum(cos(2*n)/2^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
cos⁡(2n)2n\frac{\cos{\left(2 n \right)}}{2^{n}}
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⁡(2n)a_{n} = \cos{\left(2 n \right)}
and
x0=−2x_{0} = -2
,
d=−1d = -1
,
c=0c = 0
then
1R=∞~(−2+lim⁡n→∞∣cos⁡(2n)cos⁡(2n+2)∣)\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{n \to \infty} \left|{\frac{\cos{\left(2 n \right)}}{\cos{\left(2 n + 2 \right)}}}\right|\right)
Let's take the limit
we find
1R=∞~(−2+lim⁡n→∞∣cos⁡(2n)cos⁡(2n+2)∣)\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{n \to \infty} \left|{\frac{\cos{\left(2 n \right)}}{\cos{\left(2 n + 2 \right)}}}\right|\right)
R=0(−2+lim⁡n→∞∣cos⁡(2n)cos⁡(2n+2)∣)−1R = 0 \left(-2 + \lim_{n \to \infty} \left|{\frac{\cos{\left(2 n \right)}}{\cos{\left(2 n + 2 \right)}}}\right|\right)^{-1}
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.5-0.4-0.2
The answer [src] 
  oo              
 ___              
 \  `             
  \    -n         
  /   2  *cos(2*n)
 /__,             
n = 1
∑n=1∞2−ncos⁡(2n)\sum_{n=1}^{\infty} 2^{- n} \cos{\left(2 n \right)} 
Sum(2^(-n)*cos(2*n), (n, 1, oo))
Numerical answer [src] 
-0.274929801038943632397576981589
-0.274929801038943632397576981589
The graph
Sum of series cos(2n)/(2^n)

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