Mister Exam

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Sum of series:
sqrt(n+1)-sqrt(n)/sqrt(n^2+n)
factorial(n+1)/8^(n+1)
(3*x^2-2*x+3)/factorial(x+1)
x*a^x
Identical expressions
factorial(n+ one)/ eight ^(n+ one)
factorial(n plus 1) divide by 8 to the power of (n plus 1)
factorial(n plus one) divide by eight to the power of (n plus one)
factorial(n+1)/8(n+1)
factorialn+1/8n+1
factorialn+1/8^n+1
factorial(n+1) divide by 8^(n+1)
Similar expressions
factorial(n+1)/8^(n-1)
factorial(n-1)/8^(n+1)

Sum of series/ factorial(n+1)/8^(n+1)

Sum of series factorial(n+1)/8^(n+1)

The solution

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  oo          
____          
\   `         
 \    (n + 1)!
  \   --------
  /     n + 1 
 /     8      
/___,         
n = 1

$$\sum_{n=1}^{\infty} \frac{\left(n + 1\right)!}{8^{n + 1}}$$

Sum(factorial(n + 1)/8^(n + 1), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{\left(n + 1\right)!}{8^{n + 1}}$$
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
$$a_{n} = 8^{- n - 1} \left(n + 1\right)!$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(8^{- n - 1} \cdot 8^{n + 2} \left|{\frac{\left(n + 1\right)!}{\left(n + 2\right)!}}\right|\right)$$
Let's take the limit
we find

False

False

The rate of convergence of the power series

The answer [src]

  oo                  
 ___                  
 \  `                 
  \    -1 - n         
  /   8      *(1 + n)!
 /__,                 
n = 1

$$\sum_{n=1}^{\infty} 8^{- n - 1} \left(n + 1\right)!$$

Sum(8^(-1 - n)*factorial(1 + 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)/8^(n+1)

The solution

Examples of finding the sum of a series