Mister Exam

# Sum of series (n+1)/factorial(n)

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### The solution

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

False
The rate of convergence of the power series
The answer [src]
-1 + 2*E
$$-1 + 2 e$$
-1 + 2*E
Numerical answer [src]
4.43656365691809047072057494271
4.43656365691809047072057494271`
The graph
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