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

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


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The solution

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