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(1/sqrt(factorial(n)))^2

  • How to use it?

  • Sum of series:
  • (n+1)^2/2^(n-1) (n+1)^2/2^(n-1)
  • (-1)^n/n (-1)^n/n
  • (5/6)^n (5/6)^n
  • 28 28

  • Identical expressions

  • (one /sqrt(factorial(n)))^ two
  • (1 divide by square root of (factorial(n))) squared
  • (one divide by square root of (factorial(n))) to the power of two
  • (1/√(factorial(n)))^2
  • (1/sqrt(factorial(n)))2
  • 1/sqrtfactorialn2
  • (1/sqrt(factorial(n)))²
  • (1/sqrt(factorial(n))) to the power of 2
  • 1/sqrtfactorialn^2
  • (1 divide by sqrt(factorial(n)))^2
Sum of series/ (1/sqrt(factorial(n)))^2

Sum of series (1/sqrt(factorial(n)))^2


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The solution

You have entered [src] 
  oo           
____           
\   `          
 \            2
  \   /  1   \ 
   )  |------| 
  /   |  ____| 
 /    \\/ n! / 
/___,          
n = 1
$$\sum_{n=1}^{\infty} \left(\frac{1}{\sqrt{n!}}\right)^{2}$$ 
Sum((1/(sqrt(factorial(n))))^2, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{1}{\sqrt{n!}}\right)^{2}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{1}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
The answer [src] 
-1 + E
$$-1 + e$$ 
-1 + E
Numerical answer [src] 
1.71828182845904523536028747135
1.71828182845904523536028747135
The graph
Sum of series (1/sqrt(factorial(n)))^2

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