Mister Exam

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Sum of series:
((n^3+3n+1)^(1/2))arcsin(2/n^3)
3/((n+2)^2-1)
tan(pi/(4*n))
arctg(n)/(1+n^2)
Factor polynomial:
n^3-n^2
Identical expressions
n^ three -n^ two
n cubed minus n squared
n to the power of three minus n to the power of two
n3-n2
n³-n²
n to the power of 3-n to the power of 2
Similar expressions
n^3+n^2

Sum of series/ n^3-n^2

Sum of series n^3-n^2

The solution

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  oo           
 ___           
 \  `          
  \   / 3    2\
  /   \n  - n /
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n = 0

$$\sum_{n=0}^{\infty} \left(n^{3} - n^{2}\right)$$

Sum(n^3 - n^2, (n, 0, oo))

The radius of convergence of the power series

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

True

False

The rate of convergence of the power series

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 n^3-n^2

The solution

Examples of finding the sum of a series