Mister Exam

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Sum of series:
a^n
(-1)^n*(2*2)^(2*n)/factorial(2*n)
log(n+1)/(n+1)
38
Identical expressions
one /(n*(n- two)*(n- three))
1 divide by (n multiply by (n minus 2) multiply by (n minus 3))
one divide by (n multiply by (n minus two) multiply by (n minus three))
1/(n(n-2)(n-3))
1/nn-2n-3
1 divide by (n*(n-2)*(n-3))
Similar expressions
1/(n*(n-2)*(n+3))
1/(n*(n+2)*(n-3))

Sum of series/ 1/(n*(n-2)*(n-3))

Sum of series 1/(n(n-2)(n-3))

The solution

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  oo                   
 ___                   
 \  `                  
  \           1        
   )  -----------------
  /   n*(n - 2)*(n - 3)
 /__,                  
n = 1

$$\sum_{n=1}^{\infty} \frac{1}{n \left(n - 2\right) \left(n - 3\right)}$$

Sum(1/((n*(n - 2))*(n - 3)), (n, 1, oo))

The radius of convergence of the power series

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

True

False

The rate of convergence of the power series

The answer [src]

  oo                     
 ___                     
 \  `                    
  \            1         
   )  -------------------
  /   n*(-3 + n)*(-2 + n)
 /__,                    
n = 1

$$\sum_{n=1}^{\infty} \frac{1}{n \left(n - 3\right) \left(n - 2\right)}$$

Sum(1/(n*(-3 + n)*(-2 + n)), (n, 1, oo))

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-2)*(n-3))

The solution

Examples of finding the sum of a series

Sum of series 1/(n(n-2)(n-3))