Mister Exam

### Other calculators

• #### How to use it?

• Sum of series:
• (2/9)^n
• n^2/n!
• (2/3)^n
• (-1)^(n+1)/n
• #### Identical expressions

• n^ two /n!
• n squared divide by n!
• n to the power of two divide by n!
• n2/n!
• n²/n!
• n to the power of 2/n!
• n^2 divide by n!

# Sum of series n^2/n!

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### The solution

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

False
The rate of convergence of the power series
2*E
$$2 e$$
2*E
5.43656365691809047072057494271
5.43656365691809047072057494271`