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Sum of series/ (1000^n)/n!

Sum of series (1000^n)/n!

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  oo       
____       
\   `      
 \        n
  \   1000 
  /   -----
 /      n! 
/___,      
n = 1

\sum_{n=1}^{\infty} \frac{1000^{n}}{n!}

Sum(1000^n/factorial(n), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{1000^{n}}{n!}

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} = -1000

d = 1

c = 0

then

R = \tilde{\infty} \left(-1000 + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|\right)

R = \infty

The rate of convergence of the power series

The answer [src]

      1000
-1 + e

-1 + e^{1000}

-1 + exp(1000)

Numerical answer [src]

0.e+434

0.e+434

The graph