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n+1/n(n+3)
Sum of series/ n+1/n(n+3)

Sum of series n+1/n(n+3)


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The solution

You have entered [src] 
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  \   /    n + 3\
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n = 1
$$\sum_{n=1}^{\infty} \left(n + \frac{n + 3}{n}\right)$$ 
Sum(n + (n + 3)/n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$n + \frac{n + 3}{n}$$
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
$$a_{n} = n + \frac{n + 3}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n + \frac{n + 3}{n}}{n + 1 + \frac{n + 4}{n + 1}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
oo
$$\infty$$ 
oo
Numerical answer
The series diverges
The graph
Sum of series n+1/n(n+3)

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