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Sum of series/ ((3*x)^n)/n!

Sum of series ((3*x)^n)/n!


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The solution

You have entered [src] 
  oo        
____        
\   `       
 \         n
  \   (3*x) 
  /   ------
 /      n!  
/___,       
n = 2
$$\sum_{n=2}^{\infty} \frac{\left(3 x\right)^{n}}{n!}$$ 
Sum((3*x)^n/factorial(n), (n, 2, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(3 x\right)^{n}}{n!}$$
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
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 3$$
then
$$R = \frac{\lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|}{3}$$
Let's take the limit
we find
$$R = \infty$$
The answer [src] 
     /  2   2*x         \
     |- - - ---      3*x|
   2 |  9    3    2*e   |
9*x *|--------- + ------|
     |     2          2 |
     \    x        9*x  /
-------------------------
            2
$$\frac{9 x^{2} \left(\frac{- \frac{2 x}{3} - \frac{2}{9}}{x^{2}} + \frac{2 e^{3 x}}{9 x^{2}}\right)}{2}$$ 
9*x^2*((-2/9 - 2*x/3)/x^2 + 2*exp(3*x)/(9*x^2))/2

    Examples of finding the sum of a series

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