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Sum of series:
x^(n-1)
ln^n(x)/n!
1/2
1/(n(n+5))
Identical expressions
one /(n(n+ five))
1 divide by (n(n plus 5))
one divide by (n(n plus five))
1/nn+5
1 divide by (n(n+5))
Similar expressions
1/n(n+5)
1/n*(n+5)
1/((n)*(n+5))
1/(n(n-5))
1/(n)*(n+5)
1/(n*(n+5))

Sum of series/ 1/(n(n+5))

Sum of series 1/(n(n+5))

The solution

You have entered [src]

  oo           
 ___           
 \  `          
  \       1    
   )  ---------
  /   n*(n + 5)
 /__,          
n = 1

$$\sum_{n=1}^{\infty} \frac{1}{n \left(n + 5\right)}$$

Sum(1/(n*(n + 5)), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{1}{n \left(n + 5\right)}$$
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
$$a_{n} = \frac{1}{n \left(n + 5\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(n + 6\right)}{n \left(n + 5\right)}\right)$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

            0                 0  
   -8 + 20*e        -52 + 37*e   
--------------- + ---------------
  /          0\     /          0\
4*\-60 + 60*e /   5*\-60 + 60*e /

$$\frac{-52 + 37 e^{0}}{5 \left(-60 + 60 e^{0}\right)} + \frac{-8 + 20 e^{0}}{4 \left(-60 + 60 e^{0}\right)}$$

(-8 + 20*exp_polar(0))/(4*(-60 + 60*exp_polar(0))) + (-52 + 37*exp_polar(0))/(5*(-60 + 60*exp_polar(0)))

Numerical answer [src]

0.456666666666666666666666666667

0.456666666666666666666666666667

The graph

Examples of finding the sum of a series

The Sum of the Power Series
```
x^n/n
```
```
(x-1)^n
```
Factorial
```
1/2^(n!)
```
```
n^2/n!
```
```
x^n/n!
```
```
k!/(n!*(n+k)!)
```
Flint Hills Series
```
csc(n)^2/n^3
```
Basel problem
```
1/n^2
```
```
1/n^4
```
```
1/n^6
```
Harmonic series
```
1/n
```
Grandi's series
```
(-1)^n
```
Alternating series
```
(-1)^(n + 1)/n
```
```
(n + 2)*(-1)^(n - 1)
```
```
(3*n - 1)/(-5)^n
```
```
(-1)^(n - 1)*n/(6*n - 5)
```
Newton–Mercator series
```
(-1)^(n + 1)/n*x^n
```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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How to use it?

Identical expressions

Similar expressions

Sum of series 1/(n(n+5))

The solution

Examples of finding the sum of a series