Mister Exam

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Sum of series:
(3^n-1)/6^n
1/(3n+2)(3n+1)
(factorial(n-1)/factorial(n+1))^(n*(n+1))
cos(2n)/n^2
Identical expressions
factorial(n+ one)/((seven ^n*n))
factorial(n plus 1) divide by ((7 to the power of n multiply by n))
factorial(n plus one) divide by ((seven to the power of n multiply by n))
factorial(n+1)/((7n*n))
factorialn+1/7n*n
factorial(n+1)/((7^nn))
factorial(n+1)/((7nn))
factorialn+1/7nn
factorialn+1/7^nn
factorial(n+1) divide by ((7^n*n))
Similar expressions
factorial(n-1)/((7^n*n))

Sum of series/ factorial(n+1)/((7^n*n))

Sum of series factorial(n+1)/((7^n*n))

The solution

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  oo          
____          
\   `         
 \    (n + 1)!
  \   --------
  /      n    
 /      7 *n  
/___,         
n = 1

$$\sum_{n=1}^{\infty} \frac{\left(n + 1\right)!}{7^{n} n}$$

Sum(factorial(n + 1)/((7^n*n)), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{\left(n + 1\right)!}{7^{n} n}$$
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
$$a_{n} = \frac{\left(n + 1\right)!}{n}$$
and
$$x_{0} = -7$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-7 + \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left|{\frac{\left(n + 1\right)!}{\left(n + 2\right)!}}\right|}{n}\right)\right)$$
Let's take the limit
we find

False

$$R = 0$$

The rate of convergence of the power series

The answer [src]

  oo              
____              
\   `             
 \     -n         
  \   7  *(1 + n)!
  /   ------------
 /         n      
/___,             
n = 1

$$\sum_{n=1}^{\infty} \frac{7^{- n} \left(n + 1\right)!}{n}$$

Sum(7^(-n)*factorial(1 + n)/n, (n, 1, oo))

Numerical answer

The series diverges

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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Identical expressions

Similar expressions

Sum of series factorial(n+1)/((7^n*n))

The solution

Examples of finding the sum of a series