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Sum of series/ 1/(x(x+2)(x+3)(x+1))

Sum of series 1/(x(x+2)(x+3)(x+1))


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The solution

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  oo                           
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  \               1            
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  /   x*(x + 2)*(x + 3)*(x + 1)
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n = 1
$$\sum_{n=1}^{\infty} \frac{1}{x \left(x + 2\right) \left(x + 3\right) \left(x + 1\right)}$$ 
Sum(1/(((x*(x + 2))*(x + 3))*(x + 1)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{x \left(x + 2\right) \left(x + 3\right) \left(x + 1\right)}$$
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
$$a_{n} = \frac{1}{x \left(x + 1\right) \left(x + 2\right) \left(x + 3\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} 1$$
Let's take the limit
we find
True

False
The answer [src] 
            oo           
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x*(1 + x)*(2 + x)*(3 + x)
$$\frac{\infty}{x \left(x + 1\right) \left(x + 2\right) \left(x + 3\right)}$$ 
oo/(x*(1 + x)*(2 + x)*(3 + x))

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