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pi^(2*n)/factorial(2*n)

  • How to use it?

  • Sum of series:
  • ln2 ln2
  • exp(-n) exp(-n)
  • 1/((3*n-2)(3*n+1)) 1/((3*n-2)(3*n+1))
  • pi^(2*n)/factorial(2*n) pi^(2*n)/factorial(2*n)

  • Identical expressions

  • pi^(two *n)/factorial(two *n)
  • Pi to the power of (2 multiply by n) divide by factorial(2 multiply by n)
  • Pi to the power of (two multiply by n) divide by factorial(two multiply by n)
  • pi(2*n)/factorial(2*n)
  • pi2*n/factorial2*n
  • pi^(2n)/factorial(2n)
  • pi(2n)/factorial(2n)
  • pi2n/factorial2n
  • pi^2n/factorial2n
  • pi^(2*n) divide by factorial(2*n)
Sum of series/ pi^(2*n)/factorial(2*n)

Sum of series pi^(2*n)/factorial(2*n)


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The solution

You have entered [src] 
  oo        
____        
\   `       
 \      2*n 
  \   pi    
  /   ------
 /    (2*n)!
/___,       
n = 1
∑n=1∞π2n(2n)!\sum_{n=1}^{\infty} \frac{\pi^{2 n}}{\left(2 n\right)!} 
Sum(pi^(2*n)/factorial(2*n), (n, 1, oo))
The radius of convergence of the power series
Given number:
π2n(2n)!\frac{\pi^{2 n}}{\left(2 n\right)!}
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=1(2n)!a_{n} = \frac{1}{\left(2 n\right)!}
and
x0=−πx_{0} = - \pi
,
d=2d = 2
,
c=0c = 0
then
R2=∞~(−π+lim⁡n→∞∣(2n+2)!(2n)!∣)R^{2} = \tilde{\infty} \left(- \pi + \lim_{n \to \infty} \left|{\frac{\left(2 n + 2\right)!}{\left(2 n\right)!}}\right|\right)
Let's take the limit
we find
R2=∞R^{2} = \infty
R=∞R = \infty
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.5015
The answer [src] 
  2 /   2    2*cosh(pi)\
pi *|- --- + ----------|
    |    2        2    |
    \  pi       pi     /
------------------------
           2
π2(−2π2+2cosh⁡(π)π2)2\frac{\pi^{2} \left(- \frac{2}{\pi^{2}} + \frac{2 \cosh{\left(\pi \right)}}{\pi^{2}}\right)}{2} 
pi^2*(-2/pi^2 + 2*cosh(pi)/pi^2)/2
Numerical answer [src] 
10.5919532755215206277517520526
10.5919532755215206277517520526
The graph
Sum of series pi^(2*n)/factorial(2*n)

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