Sum of series 1/(n(n+1)(n+2))

The solution

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  oo                   
 ___                   
 \  `                  
  \           1        
   )  -----------------
  /   n*(n + 1)*(n + 2)
 /__,                  
n = 1

$$\sum_{n=1}^{\infty} \frac{1}{n \left(n + 1\right) \left(n + 2\right)}$$

Sum(1/((n*(n + 1))*(n + 2)), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{1}{n \left(n + 1\right) \left(n + 2\right)}$$
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
$$a_{n} = \frac{1}{n \left(n + 1\right) \left(n + 2\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n + 3}{n}\right)$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

1/4

$$\frac{1}{4}$$

1/4

Numerical answer [src]

0.250000000000000000000000000000

0.250000000000000000000000000000

The graph

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```
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```
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```
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```
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```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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Sum of series 1/(n(n+1)(n+2))

The solution

Examples of finding the sum of a series