Mister Exam

Other calculators

Sum of series/ factorial(k)/((factorial(n)*factorial(n+k)))

Sum of series factorial(k)/((factorial(n)*factorial(n+k)))


∑
=

The solution

You have entered [src] 
  oo             
 ___             
 \  `            
  \        k!    
   )  -----------
  /   n!*(n + k)!
 /__,            
n = 1
$$\sum_{n=1}^{\infty} \frac{k!}{n! \left(k + n\right)!}$$ 
Sum(factorial(k)/((factorial(n)*factorial(n + k))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{k!}{n! \left(k + n\right)!}$$
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
$$a_{n} = \frac{k!}{n! \left(k + n\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)! \left(k + n + 1\right)!}{n! \left(k + n\right)!}}\right|$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)! \left(k + n + 1\right)!}{n! \left(k + n\right)!}}\right|$$
False
The answer [src] 
(-1 - k + (1 + k)*besseli(k, 2)*Gamma(1 + k))*k!
------------------------------------------------
                  Gamma(2 + k)
$$\frac{\left(- k + \left(k + 1\right) I_{k}\left(2\right) \Gamma\left(k + 1\right) - 1\right) k!}{\Gamma\left(k + 2\right)}$$ 
(-1 - k + (1 + k)*besseli(k, 2)*gamma(1 + k))*factorial(k)/gamma(2 + k)

    Examples of finding the sum of a series

    © Mister Exam – Calculator