Mister Exam

Lang:

Sum of series a^n/factorial(n)

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  oo    
____    
\   `   
 \     n
  \   a 
  /   --
 /    n!
/___,   
n = 0

\sum_{n=0}^{\infty} \frac{a^{n}}{n!}

Sum(a^n/factorial(n), (n, 0, oo))

The radius of convergence of the power series

Given number:

\frac{a^{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} = - a

d = 1

c = 0

then

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

R = \tilde{\infty} \left(- a + \infty\right)

The answer [src]

a
e

e^{a}

exp(a)