Mister Exam

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Sum of series n^n/factorial(n+3)

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  oo          
____          
\   `         
 \        n   
  \      n    
  /   --------
 /    (n + 3)!
/___,         
n = 1

\sum_{n=1}^{\infty} \frac{n^{n}}{\left(n + 3\right)!}

Sum(n^n/factorial(n + 3), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{n^{n}}{\left(n + 3\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{n^{n}}{\left(n + 3\right)!}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(n^{n} \left(n + 1\right)^{- n - 1} \left|{\frac{\left(n + 4\right)!}{\left(n + 3\right)!}}\right|\right)

False

False

False

The rate of convergence of the power series

The graph