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1/(2*n+1)!
Sum of series/ 1/(2*n+1)!

Sum of series 1/(2*n+1)!


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The solution

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  oo            
 ___            
 \  `           
  \       1     
   )  ----------
  /   (2*n + 1)!
 /__,           
n = 1
∑n=1∞1(2n+1)!\sum_{n=1}^{\infty} \frac{1}{\left(2 n + 1\right)!} 
Sum(1/factorial(2*n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
1(2n+1)!\frac{1}{\left(2 n + 1\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+1)!a_{n} = \frac{1}{\left(2 n + 1\right)!}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞∣(2n+3)!(2n+1)!∣1 = \lim_{n \to \infty} \left|{\frac{\left(2 n + 3\right)!}{\left(2 n + 1\right)!}}\right|
Let's take the limit
we find
False

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50.1600.180
The answer [src] 
-1 + sinh(1)
−1+sinh⁡(1)-1 + \sinh{\left(1 \right)} 
-1 + sinh(1)
Numerical answer [src] 
0.175201193643801456882381850596
0.175201193643801456882381850596
The graph
Sum of series 1/(2*n+1)!

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