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  • Sum of series:
  • n^(2/3)*arctg(1/n^2) n^(2/3)*arctg(1/n^2)
  • 2n 2n
  • 2/(n^2+6n+8) 2/(n^2+6n+8)
  • log(1+3/n) log(1+3/n)

  • Identical expressions

  • ((- one)^(n*pi^2n))/2n!
  • (( minus 1) to the power of (n multiply by Pi squared n)) divide by 2n!
  • (( minus one) to the power of (n multiply by Pi squared n)) divide by 2n!
  • ((-1)(n*pi2n))/2n!
  • -1n*pi2n/2n!
  • ((-1)^(n*pi²n))/2n!
  • ((-1) to the power of (n*pi to the power of 2n))/2n!
  • ((-1)^(npi^2n))/2n!
  • ((-1)(npi2n))/2n!
  • -1npi2n/2n!
  • -1^npi^2n/2n!
  • ((-1)^(n*pi^2n)) divide by 2n!

  • Similar expressions

  • ((1)^(n*pi^2n))/2n!
Sum of series/ ((-1)^(n*pi^2n))/2n!

Sum of series ((-1)^(n*pi^2n))/2n!


∑
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The solution

You have entered [src] 
  oo             
____             
\   `            
 \            2  
  \       n*pi *n
   )  (-1)       
  /   -----------
 /       (2*n)!  
/___,            
n = 1
∑n=1∞(−1)nπ2n(2n)!\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n \pi^{2} n}}{\left(2 n\right)!} 
Sum((-1)^((n*pi^2)*n)/factorial(2*n), (n, 1, oo))
The radius of convergence of the power series
Given number:
(−1)nπ2n(2n)!\frac{\left(-1\right)^{n \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)π2n2(2n)!a_{n} = \frac{\left(-1\right)^{\pi^{2} n^{2}}}{\left(2 n\right)!}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞∣(2n+2)!(2n)!∣1 = \lim_{n \to \infty} \left|{\frac{\left(2 n + 2\right)!}{\left(2 n\right)!}}\right|
Let's take the limit
we find
False

False
The answer [src] 
  oo            
____            
\   `           
 \          2  2
  \       pi *n 
   )  (-1)      
  /   ----------
 /      (2*n)!  
/___,           
n = 1
∑n=1∞(−1)π2n2(2n)!\sum_{n=1}^{\infty} \frac{\left(-1\right)^{\pi^{2} n^{2}}}{\left(2 n\right)!} 
Sum((-1)^(pi^2*n^2)/factorial(2*n), (n, 1, oo))
Numerical answer [src] 
0.454643416582194656423273175151 - 0.240001132562733326845290654056*i
0.454643416582194656423273175151 - 0.240001132562733326845290654056*i

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