Mister Exam

Other calculators

Sum of series/ a^n/factorial(n)

Sum of series a^n/factorial(n)


∑
=

The solution

You have entered [src] 
  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
$$R = \tilde{\infty} \left(- a + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|\right)$$
Let's take the limit
we find
$$R = \tilde{\infty} \left(- a + \infty\right)$$
The answer [src] 
 a
e
$$e^{a}$$ 
exp(a)

    Examples of finding the sum of a series

    © Mister Exam – Calculator